Hilbert spaces expanded with a unitary operator. - Departamento de ...
HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR CAMILO ARGOTY AND ALEXANDER BERENSTEIN
Abstract. We study Hilbert spaces expanded with a unitary operator with a countable spectrum. We show the theory of such a structure is ω-stable and has quantifier elimination.
1. Introduction This paper deals with the expansion of a Hilbert space with a unitary operator with a countable spectrum from the perpective of continuous logic ( for an introduction to this subject see [4]). We study the model-theoretic properties of such a structure in terms of the spectrum of the operator. In other words, our goal is to study the relation between Model Theory and Spectral Theory when the spectrum of the operator involved is countable. The main results are the following: 1.1. Theorem. Let U be a unitary operator with pure point spectrum in a Hilbert space H. Then, the theory of structure (H, U ) = (H, +, 0, h|i, U ) admits quantifier elimination, is separably categorical (see Definition 3.18), is ω-stable (see Definition 3.20) and the projection into every eigenspace is definable. 1.2. Theorem. Let U be a unitary operator whose spectrum has countably many non-empty accumulation points. Then, the theory of the structure (H, U ) = (H, +, 0, h|i, U ) admits quantifier elimination and is ω-stable. The projection into an eigenspace is definable if and only if the corresponding eigenvalue is an isolated point in the spectrum. The theory of the structure (H, +, 0, h|i, U ) where U is a unitary operator, was first studied by Henson and Iovino in [11], where they observed that it is stable. A geometric characterization of forking in such structures was first done by Berenstein and Buechler [7] after adding to the structure the projections corresponding to the The authors would like to thank Ward Henson for allowing us to include in this paper an unpublished result of his. 1
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
Spectral Decomposition Theorem. Ben Yaacov, Usvyatsov and Zadka characterized the unitary operators corresponding to generic automorphisms of a Hilbert space as those unitary transformations whose spectrum is S 1 . In this work we study the definable aspects of the spectral decomposition, we classify the separable models of the expansions and we study orthogonality of types when U has a countable spectrum. We also give a different proof of the characterization of forking given in [7]: we provide an explicit freeness relation and prove some of its properties which show it coincides with non-forking, without adding to the structure the projections corresponding to the spectral decomposition. The framework for this work is continuous logic, we assume the reader is familiar with notions such as definability, definable and algebraic closure, and forking. The background can be found in [4, 5]. This paper is divided as follows. In the second section we give a brief introduction to Spectral Theory. In the third section we study the expansions of a Hilbert space with unitary operators with a pure point (finite) spectrum and with a countably infinite spectrum.
2. Preliminaries: Spectral theory The following section is based on [1] and [12]. Let H = (H, +, 0, h|i) be an infinite dimensional Hilbert space over C. 2.1. Definition. Let A be a linear operator from H into H. The operator A is called bounded if the set {kA(u)k : u ∈ H, kuk = 1} is bounded in R. If A is bounded we define the norm of A by: kAk =
sup
kA(u)k
u∈H,kuk=1
2.2. Definition. A sequence of linear operators {An }n∈ω converges uniformly to an operator A, if limn→∞ kA − An k = 0. 2.3. Definition. Given a linear operator A : H → H, its adjoint operator, denoted A∗ is the linear operator A∗ : H → H such that for every u, v ∈ H, hAu|vi = hu|A∗ vi. 2.4. Definition. A linear operator A : H → H is called self-adjoint if A = A∗ .
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2.5. Definition. Let N a linear operator from H to H. N is called normal if N commutes with its adjoint N ∗ . 2.6. Definition. Given a normal operator A, a complex number λ is called an eigenvalue or punctual spectral value of A if the operator A − λI is not one to one. A complex number λ is called a continuous spectral value if the operator A − λI is one to one and the operator (A − λI)−1 is densely defined but is unbounded. 2.7. Fact (Theorem 1, section 93 in [1]). A point r belongs to the continuous spectrum of an operator A if and only if there is an orthonormal sequence of elements (fn : n ∈ ω) such that limn→∞ (Afn − rfn ) = 0. 2.8. Definition. The spectrum of an operator A, denoted by σ(A), is the set of the punctual and continuous spectral values. It is well known that if A is a bounded normal operator, then σ(A) is a bounded subset of C and that if A is self-adjoint, then σ(A) is a subset of R. 2.9. Definition. A self-adjoint operator A different from the zero operator is called positive and we write A ≥ 0, if hAu|ui ≥ 0 for all u ∈ H. If A and B are selfadjoint operators, we write A ≥ B if A − B is positive. 2.10. Proposition. For a self-adjoint operator A, A2 is positive. Proof. Clear.
2.11. Definition. A self-adjoint Q operator is called a square root of a positive self-adjoint operator A if Q2 = A. 2.12. Fact (Theorem 1, section 36 in [12]). Let A be a self-adjoint operator and let E+ be the projection of H onto the null space of the operator A − Q where Q is the positive square root of A2 . Then (1) Any bounded operator R that commutes with A, commutes with E+ . (2) AE+ ≥ 0, A(Id − E+ ) ≤ 0. 2.13. Fact (Theorem 2, section 36 in [12]). Let A be a bounded self-adjoint operator, let r be a real number and let E+ (r) the projection operator constructed for Ar = A−rI according to Fact 2.12. If we denote by Er the projection operator Id−E+ (r), then the family {Er }r∈R satisfies the following conditions:
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
(1) Any bounded operator R that commutes with A commutes with Er . (2) Er ≤ Es if r < s. (3) Er is continuous on the left: ∪s 0, A = m rdEr , where the integral is to be interpreted as limit of finite sums in the sense of uniform convergence in the space of operators, and m = inf(σ(A)) and M = sup(σ(A)). Let A as above. The Spectral Decomposition Theorem can be interpreted in the following way: given > 0 and δ > 0 there is an N such that for any partition m = r0 < s0 = r1 < s1 = r2 < · · · < sN = M + with max{sk − rk } < 2(M + )/N , PN if we write E(∆k ) for Esk − Erk , we have kA − k1 rk E(∆k )k < δ. 2.16. Definition. Let A be a normal operator. The real and imaginary parts of A are the operators Ar = 21 (A + A∗ ) and Ai = 12 (A − A∗ ) 2.17. Fact. The real and imaginary parts Ar and Ai of a normal operator A are self-adjoint. Proof. For u, v ∈ H, hAr u|vi = h 12 (A + A∗ )u|vi = hu| 12 (A∗ + A∗∗ iv = hu| 21 (A + A∗ )vi = hu|Ar vi. Similarly for Ai .
2.18. Corollary. If A is a normal operator, then Ar and Ai have integral repreRb Rb sentations Ar = a sdEs and Ai = a tdFt , where Es Ft = Ft Es , and we can write RR A= (s + it)dEs dFt . 2.19. Definition. A normal operator U on H is said to be unitary if U is a bijection and for any u, v ∈ H, hU u|U vi = hu|vi. 2.20. Fact (Section 71 in [1]). Let U be a unitary operator. Then there is a unique Rπ resolution of the identity {Er |r ∈ [0, 2π]} such that U k = 0 eikt dEt for all k ∈ Z.
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We want to know if we can recover 2.21. Theorem (Theorem 2.6.3 in [3]). Let N be a normal operator on H, let B(σ(N )) the algebra of Borel functions from σ(N ) into the complex numbers, and let B(H) the algebra of linear operators on H. Then there exist a isometric monomorphism π : B(σ(N )) → B(H) such that π(f¯) = (π(f )∗ ), π(1) = Id and if f = P P P P i j ¯ , then π(f ) = i j aij N i (N ∗ )j , where by 1 we denote the constant i j aij z z function on σ(N ) with value 1. 2.22. Fact. Let A be a selfadjoint operator on H. Let m = inf{σ(A)} and M = sup(σ(A)). Let f the function on [m, M ] such that f (x) = 0 if x ≤ 0 and f (x) = 1 if x > 0. Then f is aproximable by polynomials. Proof. This function clearly belongs to L2 [m, M ] and the polynomials are dense in L2 [m.M ], so the conclusion follows.
2.23. Theorem. Let A be a selfadjoint operator. Let E+ be the projection of H onto the null space of the operator A − Q where Q is the positive square root of A2 . Then E+ is approximable by operators of the form g(A) where g is a polynomial. Proof. For any v ∈ H we have E+ (v) =
RM m
g(r)dEr (v) where g is the function
defined in Fact 2.22. But by 2.22 g(x) can approximated by polynomials, E+ can also be approximated by polynomials in A.
It is important to note that for a selfadjoint operator A on a Hilbert space H, E+ is the limit of a sequence of polynomials in A, but that the sequence does not converge uniformly. That is, given a unit vector v ∈ H, E+ (v) can be approximated by polynomials in A(v), but the rate of convergence depends on v. 2.24. Definition. The essential spectrum σe (U ) of a linear operator U is the set of accumulation points of the spectrum of U together with the set of eigenvalues of infinite multiplicity. The finite spectrum σf in (U ) of a linear operator U , the complement of σe (U ), is the set of isolated points of the spectrum of U of finite multiplicity.
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
3. Unitary operators We begin this section by characterizing the theory of the expansion of a Hilbert space with a unitary operator in terms of the spectrum of the operator using an instance of a result by C. Ward Henson: 3.1. Theorem. Let A and B be two normal operators on Hilbert spaces HA , HB respectively. Then the structures (HA , +, 0, h|i, A) and (HB , +, 0, h|i, B) are elementarily equivalent if and only if (1) σe (A) = σe (B). (2) dim{x ∈ HA : Ax = λx} = dim{x ∈ HB : Bx = λx} for λ ∈ S 1 \ σe (A). We will provide a proof for the result when the operators involved have a countable spectrum and are unitary. Henson’s original proof (unpublished), based on a Theorem by Voiculescu (see [10]), yields stronger information than our proof, in particular it shows that the theory T h(H, NA ) is separably categorical up to perturbations of the automorphism. We follow the standard way of dealing with Hilbert spaces, we write the structure as (H, +, 0, h|i) and we work inside the unit ball. We denote by L the language of Hilbert spaces and by LU the language of Hilbert spaces with a new unary function, which we denote as U . Whenever we quantify over H we mean we are quantifying over its unit ball. 3.2. Notation. For λ ∈ σ(U ), we denote by Hλ the set {x ∈ H : U x = λx}, by Pλ the projection operator onto the space Hλ . 3.3. Theorem. Let UA and UB be two unitary operators with countable spectrum on Hilbert space H. Then the structures (HA , +, 0, h|i, UA ) and (HB , +, 0, h|i, UB ) are elementarily equivalent if and only if (1) σe (UA ) = σe (UB ). (2) dim{x ∈ HA : UA x = λx} = dim{x ∈ HB : UB x = λx} for λ ∈ S 1 \σe (UA ). Proof.
⇒) We may assume the structures are separable and we will write H instead of HA , HB .
Assume that the structures (H, +, 0, h|i, UA ) and
(H, +, 0, h|i, UB ) are elementarily equivalent. Let σ A be the spectrum of UA and let σ B be the spectrum of UB . For every µ ∈ S 1 \ σ A , let η = d(µ, σ A ).
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· kU u−µuk = 0 is true for (H, +, 0, h|i, UA ) Then the statement supu ηkuk− and thus true for (H, +, 0, h|i, UB ). Therefore σ B ⊂ σ A . In a similar way we can show that σ A ⊂ σ B . For every λ ∈ σfAin whose eigenspace is of dimension mλ , we have that the statement inf
u1 u2 ···umλ
max |hui |uj i|, |kui k − 1|, |U (ui ) − λui | = 0
is valid in (H, +, 0, h|i, UA ). Thus the same statement is true for (H, +, 0, h|i, UB ). From this condition and the fact that λ is an isolated point in the spectrum it easily follows that dim{x ∈ H : UA x = λx} ≤ dim{x ∈ H : UB x = λx}. In a similar way we can prove that dim{x ∈ H : UA x = λx} ≥ dim{x ∈ H : UB x = λx}. Also, for every λ ∈ σeA and every k ≥ 1, inf
u1 u2 ···uk
max |hui |uj i|, |kui k − 1|, |U (ui ) − λui | = 0
is true for (H, +, 0h|i, UA ) and thus also true for (H, +, 0, h|i, UB ). This implies that σeA = σeB . ⇐) Conversely, assume that σeA = σeB and dim{x ∈ H : UA x = λx} = dim{x ∈ H : UB x = λx} for λ ∈ σfAin . Let σe = σeA . We may assume, exchanging (HA , UA ) and (HB , UB ) for elementary superstructures, that (HA , UA ) and f in λ λ = {x ∈ , where HA (HB , UB ) are ℵ0 -saturated. Let HA = ⊕λ∈σf in HA λ HA : UA (x) = λx}. Note that HA is the domain of a finite dimensional f in λ λ = ⊕λ∈σf in HB Hilbert subspace of HA . Let HB , where HB = {x ∈ HB : f in UB (x) = λx}. (HB , UB ) is a substructure of (HB , UB ). Then for each f in f in λ λ λ ∈ σf in , (HA , UA ) ∼ , UB ), so (HA , UA ) ∼ = (HB = (HB , UB ). We denote
this common substructure as Hf in . Let HC = HA ⊕Hf in HB , the free amalgamation of HA and HB over Hf in and let UC be the induced unitary map on HC determined by UA and UB . We will prove that (HA , UA ) (HC , UC ) and that (HB , UB ) (HC , UC ). From this we get that the structures (HA , +, 0, h|i, A) and (HB , +, 0, h|i, B) are elementarily equivalent. Claim (HA , A) (HC , C) We use the Tarski-Vaught test. Let ϕ(x1 , . . . , xn , y) be a formula and n let a ¯ = (a1 , . . . , an ) ∈ HA . Assume that inf{ϕ(HC ,UC ) (a1 , . . . , an , b) : b ∈
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
HC } = r and let > 0. Let b ∈ HC be such that ϕ(a1 , . . . , an , b) − r < . We may write b = bA + d, where bA = PHA (b), so d ⊥ HA . There exists k ≥ 1 and an L-formula ψ(x11 , . . . , xn1 , y1 ; . . . ; x1k , . . . , xnk , yk )
such that (HC , C) |= supx1 . . . supxn supy |ϕ(x1 , . . . , xn , y)−ψ(x1 , . . . , xn , y; . . . ; U k (x1 ), . . . , U k (xk ), U k (y 0. Clearly d ∈ ⊕λ∈σe HλB , so d =
P
λ∈σe
Pλ (d). Note that Pλ (d) ⊥ HA so
λ Pλ (d) ⊥ Pλ (ai ) for each i ≤ n. Since (HA , UA ) is ℵ0 -saturated, dim(HA )= λ ∞ for each λ ∈ σe , so there exists cλ ∈ HA such that kcλ k = kPλ (d)k and P cλ ⊥ Pλ (a1 ), . . . , Pλ (an ), Pλ (bA ). Let c = λ∈σe cλ and let b0 = bA + c.
Then tp(c, . . . , U k (c)/{¯ a, U (¯ a), . . . , U k (¯ a), bA }) = tp(d, . . . , U k (d)/{¯ a, U (¯ a), . . . , U k (¯ a), bA }), so tp(b, . . . , U k (b)/{¯ a, U (¯ a), . . . , U k (¯ a)}) = tp(b0 , . . . , U k (b0 )/{¯ a, U (¯ a), . . . , U k (¯ a)}), so ψ(a1 , . . . , an , b, . . . , U k (a1 ), . . . , U k (ak ), U k (b)) = ψ(a1 , . . . , an , b0 , . . . , U k (a1 ), . . . , U k (ak ), U k (b0 )) and ϕ(a1 , . . . , an , b0 ) − r < . Since was arbitrary we get inf{ϕ(HC ,UC ) (a1 , . . . , an , b) : b ∈ HC } = r = inf{ϕ(HC ,UC ) (a1 , . . . , an , b) : b0 ∈ HA } In other words, since Theorem ?? this theorem means that the operator theoretic concept of aproximate unitary equivalence and the model theoretic concept of elementary equivalence. The previous Theorem is proved in [6] when σ A = σeA = S 1 . Let U be a unitary operator in a separable Hilbert space H and σ be the corresponding spectrum. The previous result says that the theory of the structure (H, U ) = (H, +, 0, h, i, U ) determines the dimension of the eigenspace corresponding to the isolated points of the spectrum. We strengthen this observation by showing the definability of these eigenspaces. 3.4. Notation. For λ ∈ σ, we denote by Bλ the unit ball in the space Hλ . Recall that a closed set D ⊂ H is definable if the function dist(x, D) is a definable predicate. We will use the following characterization of definable sets and functions: 3.5. Fact (Proposition 9.18 of [4]). Let D ⊆ H be closed. Then D is definable in (H, U ) if and only if there is a definable predicate P : H n → [0, 1] such that
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P (x) = 0 for all x ∈ D and ∀∃δ∀x ∈ H n (P (x) ≤ δ ⇒ d(x, D) ≤ ). 3.6. Fact (Proposition 9.22 of [4]). Let (H0 , U 0 ) be κ-saturated where κ is uncountn
able and let A ⊆ H 0 have cardinality < κ. Let f : H 0 → H 0 be any function and let Gf its graph. Then the following are equivalent: (1) f is definable in (H0 , U 0 ) over A. (2) Gf is type-definable in (H0 , U 0 ) over A. 3.7. Lemma. Let λ ∈ σ be isolated, let χ = d(λ, σ \ {λ}) and let u ∈ H. If kU (u) − λuk < then ku − Pλ (u)k
0∃N ∀n ≥ Pn N, kb − k=0,λ∈σ ck,λ Pλ (ak,λ )k < . Let R(x) = kx − bk and φn (x) := kx − Pn k=0,λ∈σ ck Pλ (ak,λ )k. Then ∀ > 0∃N ∀n ≥ N ∀x(|R(x) − φn (x)| < ) and {b} is definable over A.
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Let b 6∈ E, then for some λ0 ∈ σ, Pλ0 (b) 6∈ E. Since there are infinitely many vectors in Hλ0 ∩ E ⊥ with norm kPλ0 (b − PE (b))k, there are infinitely many realizations of tp(b/A) and therefore b 6∈ dcl(A).
3.16. Lemma. Let A ⊆ H. Then acl(A) is the closed Hilbert space generated by the union of dcl(A) with all the finite dimensional subspaces Hλ with λ ∈ σf in . Proof. Every ball Bλ for λ ∈ σf in is algebraic over ∅ because the closed unit ball of a finite dimensional space is compact. Therefore dcl(A) ∪λ∈σf in Hλ ⊂ acl(A). Conversely, let E be the closure of the space generated by dcl(A) and ∪λ∈σf in Hλ . If b 6∈ E, then Pλ0 (b) − Pλ0 (PE (b)) 6= 0 for some λ0 ∈ σe . Without loss of generality the dimension of Hλ0 ∩dcl(A)⊥ is infinite and the set {b0 ∈ H : tp(b0 /A) = tp(b/A)} is unbounded, thus b 6∈ acl(A).
3.17. Proposition. Let p, q ∈ S1 (∅) and let a |= p, b |= q. Then d(p, q) = pP 2 λ∈σ (kPλ (a)k − kPλ (b)k) . Proof.
00
≥00 . It is easy to see that: p
kak2 − 2ha|bi + kbk2 = sX = (kPλ (a)k2 − 2hPλ (a)|Pλ (b)i + kPλ (b)k2 )
ka − bk =
λ∈σ
≥
sX
(kPλ (a)k2 − 2kPλ (a)kkPλ (b)k + kPλ (b)k2 )
λ∈σ
=
sX
(kPλ (a)k − kPλ (b)k)2
λ∈σ
Then, d(p, q)
inf{ka0 − b0 k | H |= p(a0 ) and H |= q(b0 )} sX (kPλ (a)k − kPλ (b)k)2 . ≥ =
λ∈σ 00
≤00 . Let a0 , b0 be such that tp(a) = tp(a0 ) and tp(b) = tp(b0 ). Then kPλ (a)k =
kPλ (a0 )k and kPλ (b)k = kPλ (b0 )k for λ ∈ σ, but the inner products hPλ (a0 )|Pλ (b0 )i depend on the choice of a0 and b0 . We may choose a0 and b0 such that for each
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λ ∈ σ, hPλ (a0 )|Pλ (b0 )i = kPλ (a0 )kkPλ (b0 )k. Then, d(p, q)
=
inf{ka − bk|H |= p(a) and H |= q(b)}
ka0 − b0 k = p = ka0 k2 − 2ha0 |b0 i + kb0 k2 sX (kPλ (a0 )k2 − 2hPλ (a0 )|Pλ (b0 )i + kPλ (b0 )k2 ) = = ≤
λ∈σ
=
sX
(kPλ (a0 )k2 − 2kPλ (a0 )kkPλ (b0 )k + kPλ (b0 )k2 ) =
λ∈σ
=
sX
(kPλ (a)k − kPλ (b)k)2 .
λ∈σ
3.18. Definition. Recall that the theory of a metric structure M is called separably categorical if whenever N1 , N2 |= T h(M) are separable we have N1 ∼ = N2 . 3.19. Lemma. The theory Tσ is separably categorical. Proof. Let (H0 , U 0 ) and (H00 , U 00 ) be separable models of Tσ . Then for each λ ∈ σ, dim(Hλ0 ) = dim(Hλ00 ) (which is either finite or ℵ0 ). Hence, we have that for every λ ∈ σ, Hλ0 ∼ = Hλ00 and thus (H0 , U 0 ) ∼ = (H00 , U 00 ).
One could also prove the previous Lemma using Proposition 3.17. It is clear that the formula presented in Proposition 3.17 is definable and thus the logic topology and the distance topology agree on the space of types and by Theorem 12.4 [4] Tσ is separably categorical. 3.20. Definition. The theory of a metric structure Mis called ω-stable if for any N |= T h(M) and A ⊆ N countable, S(A) is separable. 3.21. Theorem. The theory Tσ is ω-stable. Proof. Let H |= Tσ be separable and let A ⊆ H be a countable set. Let A¯ be the L algebraic closure of A and write H = A¯ + A¯⊥ = λ∈σ Hλ . For each λ ∈ σe , let Hλ0 be a separable Hilbert space such that Hλ ∩ Hλ0 = {0} and let H 0 = H ⊕λ∈σe Hλ0 . We define for each v ∈ Hλ0 , Uλ0 (v) = λv and let U 0 be the unitary map on H0 induced by U and Uλ0 , λ ∈ σe . Then (H0 , U 0 ) |= Tσ and dim(Hλ0 ∩ A¯⊥ ) = ∞ for
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each λ ∈ σe . In particular, by Theorem 3.14 H0 realizes all types over A, so we can work inside the structure H0 . Since H0 is separable, so is S(A).
3.2. Countably infinite spectrum. Our next step is to consider a unitary operator U with countable spectrum σ. We write σ = σa ∪ σi , where σi are the isolated points of the spectrum and σa are the non-isolated points of the spectrum. 3.22. Theorem. The following properties are true: (1) Tσ has quantifier elimination. Furtheromore, for any a ¯ ∈ Hn , tp(¯ a) is determined by the values hPλ (ak ), Pλ (aj )i for λ ∈ σ, j, l ≤ n. (2) Tσ is ω-stable. Proof.
(1) It follows from Lemma ?? that for any v ∈ H and λ ∈ σ, Pλ (v) is in the quantifier free definable closure of v. The rest o the proof follows as in Theorem 3.14.
(2) We can proceed as in Theorem 3.20. 3.23. Proposition. Let p, q ∈ S1 (∅) and let a |= p, b |= q. Then d(p, q) = pP 2 λ∈σ (kPλ (a)k − kPλ (b)k) . Proof. Similar to Proposition 3.17.
3.24. Theorem. The principal types in S1 (T ) are the ones of elements a ∈ H with Pλ (a) = 0 for λ ∈ σa . Proof. We can build a structure (H0 , 0, +, h|i, U 0 ) elementarily equivalent to (H, 0, +, h|i, U ) such that Hλ = 0 for all λ ∈ σa . In this structure the types of elements a such that Pλ (a) 6= 0 for some λ ∈ σa are omitted. Conversely, assume that a is such that Pλ (a) = 0 for all λ ∈ σa and let (H0 , 0, +, h, i, U 0 ) be a model of Tσ . By Theorem 3.10 the projections Pλ are definable for every isolated λ ∈ σ and thus Hλ0 6= 0. Let a0λ ∈ Hλ0 be such that P 0 0 0 ka0λ k = kPλ (a)k for each λ ∈ σi and let a0 = λ∈σi aλ . Then a ∈ H and tp(a) = tp(a0 ). One could also use Proposition 3.23 to prove the previous Theorem.
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3.25. Corollary. The atomic models of Tσ are the models in which the accumulation points of the spectrum are not eigenvalues. 3.26. Corollary. The ℵ0 -saturated models of Tσ are the ones in which the accumulation points of the spectrum are eigenvalues whose eigenspace is infinite dimensional. 3.27. Corollary. For λ ∈ σa , the set Hλ is not definable and the function Pλ is not definable. Proof. Assume, in order to get a contradiction, that Hλ is definable. In an ℵ0 saturated structure the statement inf u∈Hλ (kuk − 1)2 = 0 is true. Thus in the prime model this property also holds and there is a vector of norm one in Hλ , a contradiction. Assume now that Pλ is definable. Then kx − Pλ (x)k measures the distance from x into Hλ (x) and thus Hλ is definable, a contradiction.
3.28. Theorem. If σa is finite then Tσ has ℵ0 nonisomorphic separable models. If σa is infinite then Tσ has 2ℵ0 nonisomorphic separable models. L Proof. Let H and H0 be separable models of Tσ . We can write H = λ∈σi ∪σa Hλ L and H0 = λ∈σi ∪σa Hλ0 . For λ ∈ σi , Hλ ∼ = Hλ0 , so the only diference between H and H0 can come from the spaces Hλ and Hλ0 for λ ∈ σa . For such λ, the spaces Hλ and Hλ0 are isomorphic if and only if dim(Hλ ) = dim(Hλ0 ). Assume first that σa is a finite non-empty set. So there is exactly one model up to isomorphism for every possible dimension of Hλ for λ ∈ σa . Thus there are ℵ0 nonisomorphic separable models of the theory Tσ . On the other hand, if σa is an infinite countable set, there are 2ℵ0 nonisomorphic separable models of Tσ .
3.3. Forking. We fix a countable spectrum σ and a structure (H, U ) |= Tσ which is κ-saturated and strongly κ-homogeneous for some uncountable inaccessible cardinal κ. We say A ⊂ H is small if |A| < κ. 3.29. Definition.
(1) Let a ¯ = (a1 , . . . , am ) ∈ H m ; let B, C ⊆ H be small, let
∗ ¯ is ∗B ∪ C = acl(B ∪ C) and C¯ = acl(C). We write a ¯^ | C B and say that a
independent from B over C if PB∪C (Pλ (ai )) = PC¯ (Pλ (ai )) for i = 1, . . . , m and λ ∈ σ. ∗
(2) For A, B, C ⊆ H small we say A ^ | C B if and only if for all finite subsets a ¯ ∗ of A, we have that a ¯^ | C B.
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3.30. Lemma. Let B be an algebraically closed small set in (H, U ). Then for every λ ∈ σ and for every a ∈ H, Pλ (PB (a)) = PB (Pλ (a)). Proof. We first need the following claim: 3.31. Claim. PB and U commute Proof. Let a ∈ H. We can write a = aB + aB ⊥ , where aB = PB (a) and aB ⊥ = a − aB . Then U (a) = U (aB ) + U (aB ⊥ ). For any b ∈ B, U (b), U ∗ (b) ∈ B. Thus hb|U (aB ⊥ )i = hU ∗ (b)|aB ⊥ i = 0. This implies that U (aB ) ∈ B and U (aB ⊥ ) ∈ B ⊥ ; PB (U (a)) = PB (U (aB ) + U (aB ⊥ )) = U (aB ) and thus PB (U (a)) = U (PB (a)).
The result follows from Fact 2.13 part 1.
3.32. Notation. For A ⊂ H small, we write A¯ for acl(A). ∗
3.33. Corollary. Let a ¯ = (a1 , . . . , am ) ∈ H m ; let B, C ⊆ H be small. Then a ¯^ | CB if and only if PB∪C (ai ) = PC (ai ) for i ≤ m. ∗
Proof. Assume first that a ¯^ | C B, then PB∪C (Pλ (ai )) = PC¯ (Pλ (ai )) for any i ≤ n. By Lemma 3.30 we get Pλ (PB∪C (ai )) = Pλ (PC¯ (ai )) for all λ and thus PB∪C (ai ) = PC¯ (ai ). The converse is proved in a similar way.
3.34. Proposition. Given two tuples a ¯ = (a1 , . . . , an ) and ¯b = (b1 , . . . , bm ) and a ∗
∗
small set C, a ¯^ | C ¯b if and only if Pλ (ak ) ^ | C Pλ (bj ) for k = 1, . . . , n; j = 1, . . . , m and λ ∈ σ. ∗ Proof. Let a ¯ = (a1 , . . . , an ) and ¯b = (b1 , . . . , bm ). If a ¯^ | C ¯b, then we have that, P{b1 ,...,bm }∪C (Pλ (ai )) = PC¯ (Pλ (ai )) for every i = 1, . . . , n and λ ∈ σ.
For λ ∈ σ and j ≤ m, we have C¯ ⊂ C ∪ {Pλ (bj )} ⊆ C ∪ {bj } ⊆ C ∪ {b1 , . . . , bm }, so for every i = 1, . . . , n, P{Pλ (bj )}∪C (Pλ (ai )) = PC¯ (Pλ (ai )). Conversely, let us suppose that P{Pλ (bj )}∪C (Pλ (ak )) = PC¯ (Pλ (ak )) for all λ ∈ σ, k = 1, . . . , n, j = 1, . . . , m. We fix λ ∈ σ. We can write Pλ (ak ) = PC¯ (Pλ (ak ))+PC¯ ⊥ (Pλ (ak )). Since P{Pλ (bj )}∪C (Pλ (ak )) = PC¯ (Pλ (ak )) for every j = 1, . . . , m, then PC¯ ⊥ (Pλ (ak )) ⊥ Pλ (bj ) for j = 1, . . . , m and PC¯ ⊥ (Pλ (ak )) ⊥ C ∪ {Pλ (bj )} for j = 1, . . . , m.
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
Thus PC¯ ⊥ (Pλ (ak )) ⊥ C ∪ {Pλ (b1 ), . . . , Pλ (bm )} and PC∪{Pλ (b1 ),...,Pλ (bm )} (Pλ (ak )) = PC¯ (Pλ (ak )). Since PC∪{b1 ,...,bn } (Pλ (ak )) belongs to the closed space Hλ , we also get PC∪{Pλ (b1 ),...,Pλ (bm )} (Pλ (ak )) = PC∪{b1 ,...,bm } (Pλ (ak )). ∗
¯^ | C ¯b. So PC∪{b1 ,...,bm } (Pλ (ai )) = PC¯ (Pλ (ai )) for any λ ∈ σ and then a
3.35. Lemma. Let C ⊆ H be algebraically closed and let a ¯ = (a1 , . . . , an ) ∈ H n , ∗ ¯b = (b1 . . . , bm ) ∈ H m be tuples in H. Then a ¯^ | C ¯b if and only if Pλ (ak ) − PC¯ (Pλ (ak )) ⊥ Pλ (bj ) for k = 1, . . . , n, j = 1, . . . , m and λ ∈ σ.
Proof. Given two tuples a ¯ = (a1 , . . . , an ) and ¯b = (b1 , . . . , bm ), by Proposition 3.34 ∗
∗
a ¯^ | C ¯b if and only if Pλ (ak ) ^ | C Pλ (bj ) for k = 1, . . . , n, j = 1, . . . , m and λ ∈ σ. This happens if and only if PC∪{Pλ (bj )} (Pλ (ak )) = PC¯ (Pλ (ak )) for k = 1 . . . , n, j = 1, . . . , m and λ ∈ σ. Finally, PC∪{Pλ (bj )} (Pλ (ak )) = PC¯ (Pλ (ak )) if and only if Pλ (ak ) − PC¯ (Pλ (ak )) ⊥ Pλ (bj ).
∗
3.36. Theorem. The relation ^ | satisfies the following properties: finite character, local character, transitivity, symmetry, invariance, existence and stationarity. Proof. We prove all the properties: ∗
∗
(1) Finite character: we show that a ¯^ | C B if and only if a ¯^ | C B0 for all finite ∗
∗
∗
B0 ⊆ B. First of all, if a ¯^ | C B and B 0 ⊆ B then a ¯^ | C B 0 . If a ¯^ 6 | C B, PB∪C (ak ) 6= PC¯ (ak ) for some 1 ≤ k ≤ n. Let b = PB∪C (ak ) − PC¯ (ak ). Then there exist b1 , . . . , bl ∈ B, c1 , . . . , cm ∈ C, λ1 , . . . , λl+m ∈ σ and Pl Pm α1 , . . . , αn , β1 , . . . , βm ∈ C such that |b− k=1 αk Pλk (bk )− j=1 βj Pλl+j (cj )| < ∗
kbk/2. Let B0 = {b1 , . . . , bl }, then a ¯^ 6 | C B0 . (2) Local Character: For every a and B there exists a sequence (dn : n ∈ ω) such that dn → PB¯ (a) and dn = α1n Pλ1n (b1n ) + · · · + αkn n Pλkn n (bkn n ) where bij ∈ B, λij ∈ σ and αij ∈ C. Let B0 = {bij : i ≤ j, j ∈ ω}. B0 ⊆ B ∗
is countable and a ^ | B B. 0 (3) Transitivity of independence: Let A ⊆ B ⊆ C ⊂ H be small and let ∗
a ¯ = (a1 , . . . , an ). If a ¯^ | A C, PC¯ (ak ) = PA¯ (ak ) for k = 1, . . . , n. So, ∗ PC¯ (ak ) = PB¯ (ak ) = PA¯ (ak ) for k = 1, . . . , n and therefore a ¯^ | B C and ∗
a ¯^ | A B. The converse is proved in a similar way. (4) Symmetry: It enough to show that for any tuples a ¯ = (a1 , . . . , an ) and ∗ ∗ ¯b = (b1 , . . . , bm ) and small sets C, a ¯^ | C ¯b if and only if ¯b ^ | Ca ¯. Let
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λ ∈ σ, k ≤ n and j ≤ m, we can write Pλ (ak ) = Pλ (akC¯ ) + Pλ (akC¯ ⊥ ) and Pλ (bj ) = Pλ (bjC¯ ) + Pλ (bjC¯ ⊥ ), where akC¯ = PC¯ (ak ), akC¯ ⊥ = PC¯ ⊥ (ak ). If ∗
a ¯^ | C ¯b by Lemma 3.35 Pλ (ak ) − Pλ (akC¯ ) ⊥ Pλ (bj ) then hPλ (akC¯ ⊥ )|Pλ (bjC¯ ) + Pλ (bjC¯ ⊥ )i = 0. So hPλ (akC¯ ⊥ )|Pλ (bjC¯ )i + hPλ (akC¯ ⊥ )|Pλ (bjC¯ ⊥ )i = 0. By Lemma 3.30 hPλ (akC¯ ⊥ )|Pλ (bjC¯ )i = 0, so hPλ (akC¯ ⊥ )|Pλ (bjC¯ ⊥ )i = 0. On the other hand, by Lemma 3.30 hPλ (akC¯ )|Pλ (bjC¯ ⊥ )i = 0, so hPλ (ak )|Pλ (bjC¯ ⊥ )i = 0. Thus hPλ (ak )|Pλ (bj ) − Pλ (bjC¯ )i = 0 and therefore Pλ (ak ) ⊥ Pλ (bj ) − Pλ (PC¯ (bj )). ∗
By Lemma 3.35, this implies that ¯b ^ | Ca ¯ which completes the proof (5) Invariance: For every u, v ∈ H and f ∈ Aut(H), hu|vi = hf (u)|f (v)i. ∗ Let a ¯ = (a1 , . . . , an ) and ¯b = (b1 , . . . , bm ). Then a ¯^ | C ¯b if and only if Pλ (ak ) − Pλ (PC¯ (ak )) ⊥ Pλ (bj ) for every k = 1, . . . , n, j = 1, . . . , m and
λ ∈ σ if and only if Pλ (f (ak )) − Pλ (Pf (C) ¯ (f (ak ))) ⊥ Pλ (f (bj )). (6) Existence: Let a ¯ = (a1 , . . . , an ) ∈ H n and let A ⊆ H be small. By quantifier elimination, the type tp(¯ a/A) is determined by PA¯ (Pλ (ak )) for λ ∈ σ and the inner products hPλ (ak )|Pλ (aj )i for k, j ≤ n. Let ¯b ∈ H n and B ⊇ A be small. tp(¯b/B) is a free extension of tp(¯ a/A) if and only if tp(¯b/A) = tp(¯ a/A) and PB¯ (Pλ (bk )) = PA¯ (Pλ (ak )) for all λ ∈ σ and k = 1, . . . , n. For each λ ∈ σ, let aλkA¯⊥ = PA¯⊥ (Pλ (ak )), aλkA¯ = PA¯ (Pλ (ak )). Since B is small, for each λ ∈ σe , dim(Hλ ∩ B ⊥ ) = ∞, so we can find P dλk ∈ Hλ ∩ B ⊥ with kaλkA¯⊥ k = kdλk k. Let bk = λ∈σ (aλkA¯ + dλk ) and let ¯b = (b1 , . . . , bn ). Then tp(¯ a/A) = tp(¯b/A) and PB¯ (Pλ (bk )) = PA¯ (Pλ (ak )) for k = 1, . . . n. (7) Stationarity: Let a ¯ = (a1 , . . . , an ), ¯b = (b1 , . . . , bn ), ¯b0 = (b01 , . . . , b0n ) ∈ H n and let A ⊆ B ⊆ H be small. Assume that the types of ¯b and ¯b0 over B are free extensions of tp(¯ a/A). Then tp(¯b/A) = tp(¯b0 /A) and for every λ ∈ σ and i ≤ n, PB¯ (Pλ (bi )) = PA¯ (Pλ (bi )) = PA¯ (Pλ (b0i )) = PB¯ (Pλ (b0i )). Thus tp(¯b/B) = tp(¯b0 /B). Therefore for every p ∈ S(A) |{tp(¯ a/B)|¯ a^ | A B, p ⊆ tp(¯ a/B)}| = 1.
3.37. Corollary. The theory Tσ is stable and ∗-independence coincides with the notion of independence induced by forking.
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
3.38. Observation. We had shown in Theorem 3.20 that Tσ is ω-stable and thus it has prime models over sets. Let A ⊂ H be small. For each λ ∈ σi such that ¯ < ℵ0 , let H 0 be a subspace of Hλ of dimension ℵ0 . dim(Hλ ) = ∞ and dim(Pλ (A)) λ Then A ⊕λ∈σi Hλ0 is the prime model over A of Tσ . It also follows from ω-stability that for every infinite cardinal µ, Tσ has a µsaturated model of dimension µ. Indeed, for each λ ∈ σe , let Hλ0 be a subspace of Hλ of dimension µ. Then ⊕Hλ0 |= Tσ has dimension µ and it is saturated. 3.39. Definition. Let A ⊆ H be small and p, q ∈ Sn (A). We say that p is almost orthogonal to q (p ⊥a q) if for all a ¯ |= p and ¯b |= q a ¯^ | A ¯b. 3.40. Definition. Let A ⊆ H and p, q ∈ Sn (A). We say that p is orthogonal to q (p ⊥ q) if for all B ⊇ A, pB ⊇ p a non-forking extension, and qB ⊇ q a non-forking extension, pB ⊥a qB . 3.41. Theorem. Let A ⊆ H be such that A = acl(A). Let p, q ∈ S1 (A), let a |= p and b |= q. Let a = PA (a) + a0 and b = PA (b) + b0 , let σp = {λ ∈ σe : Pλ (a0 ) 6= 0} σq = {λ ∈ σe : Pλ (b0 ) 6= 0}, then, p ⊥ q if and only if p ⊥a q if and only if σp ∩ σq = ∅. Proof. Assume that p ⊥ q, then p ⊥a q. Let a |= p and b |= q, let a0 = a − PA (a) and b0 = b − PA (b). The type tp(a/A) is determined by PA (a) and the norms of Pλ (a0 ) for λ ∈ σe . Assume, in order to get a contradiction, that Pλ0 (a0 ) 6= 0 and Pλ0 (b0 ) 6= 0 for some λ0 ∈ σe . Let a00 , b00 such that tp(a0 /A) = tp(a00 /A), tp(b0 /A) = tp(b00 /A) and Pλ0 (a00 ) is a multiple Pλ0 (b00 ). Then tp(PA (a) + a00 /A) = tp(a/A) and tp(PA (b) + b00 /A) = tp(b/A) but (PA (a) + a00 ) ^ 6 | A (PA (b) + b00 ), a contradiction to p ⊥w q. Conversely, assume that σp ∩ σq = ∅. Let B ⊇ A and let pB ⊃ p, qB ⊇ q be nonforking extensions. Let c and d realizations of pB and qB respectively. We may write c = cB¯ +c0 , d = dB¯ +d0 where cB¯ dB¯ are the projections of c and d over acl(B) = B. P P Then σpB = σp and σqB = σq . Then hc0 |d0 i = ( λ∈σp Pλ (a0 )| µ∈σq Pµ (b0 )i = P P 0 0 0 0 | B d. λ∈σp µ∈σq hPλ (c )|Pµ (d )i = 0. Then c ⊥ d and c ^ A generalization of the previous result appears in [6] when σf in = ∅ and A = ∅.
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References [1] N.I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert Space vols.I and II. Pitman Advanced Publishing Program, 1981. [2] Ita¨ı Ben Yaacov, On perturbations of continuous structures, submitted. [3] W. Arveson, A short course in spectral theory. Springer Verlag, 2002. [4] Ita¨ı Ben Yaacov, Alexander Berenstein, C. Ward Henson and Alexander Usvyatsov, Model theory for metric structures, to appear in the Proceedings of the Isaac Newton Institute’s semester on Model Theory and its applications. [5] Ita¨ı Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability. submitted. [6] Ita¨ı Ben Yaacov, Alexander Usvyatsov and Moshe Zadka, Generic automorphism of a Hilbert space, preprint. [7] Alexander Berenstein, and Steven Buechler, Simple stable homogeneous expansions of Hilbert spaces. Annals of of pure and Applied Logic. Vol. 128 (2004) pag 75-101. [8] Steven Buechler, Essential stability theory. Springer Verlag, 1991. [9] R. Dautray, L. Lions, Mathematical analysis and numerical methods for science and technology, volume 3. [10] Kenneth Davidson, C ∗ -Algebras by example, Field Institute Monographs, 1996. [11] Jos´ e Iovino, Stable theories in functional analysis University of Illinois Ph.D. Thesis, 1994. [12] L. Liusternik and V. Sobolev, Elements of Functional Analysis. Frederic Ungar Publishing Co., New York, 1961. [13] M. Reed, B. Simon, Methods of modern mathematicalphysics volume I:Functional analysis, revised and enlarged edition. Academic Press, 1980. [14] Werner Schmeidler, Linear Operators in Hilbert Space, Academic Press, 1965.
´ticas, Cra 1# Camilo Argoty, Universidad de los Andes, Departamento de Matema ´, Colombia. and, Universidad Sergio Arboleda, Departamento de Matema ´ticas, 18A-10, Bogota E-mail address: [email protected] ´ticas, Alexander Berenstein, Universidad de los Andes, Departamento de Matema ´, Colombia. Carrera 1 N 18A-10, Bogota E-mail address: [email protected]
Abstract. We study Hilbert spaces expanded with a unitary operator with a countable spectrum. We show the theory of such a structure is ω-stable and has quantifier elimination.
1. Introduction This paper deals with the expansion of a Hilbert space with a unitary operator with a countable spectrum from the perpective of continuous logic ( for an introduction to this subject see [4]). We study the model-theoretic properties of such a structure in terms of the spectrum of the operator. In other words, our goal is to study the relation between Model Theory and Spectral Theory when the spectrum of the operator involved is countable. The main results are the following: 1.1. Theorem. Let U be a unitary operator with pure point spectrum in a Hilbert space H. Then, the theory of structure (H, U ) = (H, +, 0, h|i, U ) admits quantifier elimination, is separably categorical (see Definition 3.18), is ω-stable (see Definition 3.20) and the projection into every eigenspace is definable. 1.2. Theorem. Let U be a unitary operator whose spectrum has countably many non-empty accumulation points. Then, the theory of the structure (H, U ) = (H, +, 0, h|i, U ) admits quantifier elimination and is ω-stable. The projection into an eigenspace is definable if and only if the corresponding eigenvalue is an isolated point in the spectrum. The theory of the structure (H, +, 0, h|i, U ) where U is a unitary operator, was first studied by Henson and Iovino in [11], where they observed that it is stable. A geometric characterization of forking in such structures was first done by Berenstein and Buechler [7] after adding to the structure the projections corresponding to the The authors would like to thank Ward Henson for allowing us to include in this paper an unpublished result of his. 1
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
Spectral Decomposition Theorem. Ben Yaacov, Usvyatsov and Zadka characterized the unitary operators corresponding to generic automorphisms of a Hilbert space as those unitary transformations whose spectrum is S 1 . In this work we study the definable aspects of the spectral decomposition, we classify the separable models of the expansions and we study orthogonality of types when U has a countable spectrum. We also give a different proof of the characterization of forking given in [7]: we provide an explicit freeness relation and prove some of its properties which show it coincides with non-forking, without adding to the structure the projections corresponding to the spectral decomposition. The framework for this work is continuous logic, we assume the reader is familiar with notions such as definability, definable and algebraic closure, and forking. The background can be found in [4, 5]. This paper is divided as follows. In the second section we give a brief introduction to Spectral Theory. In the third section we study the expansions of a Hilbert space with unitary operators with a pure point (finite) spectrum and with a countably infinite spectrum.
2. Preliminaries: Spectral theory The following section is based on [1] and [12]. Let H = (H, +, 0, h|i) be an infinite dimensional Hilbert space over C. 2.1. Definition. Let A be a linear operator from H into H. The operator A is called bounded if the set {kA(u)k : u ∈ H, kuk = 1} is bounded in R. If A is bounded we define the norm of A by: kAk =
sup
kA(u)k
u∈H,kuk=1
2.2. Definition. A sequence of linear operators {An }n∈ω converges uniformly to an operator A, if limn→∞ kA − An k = 0. 2.3. Definition. Given a linear operator A : H → H, its adjoint operator, denoted A∗ is the linear operator A∗ : H → H such that for every u, v ∈ H, hAu|vi = hu|A∗ vi. 2.4. Definition. A linear operator A : H → H is called self-adjoint if A = A∗ .
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2.5. Definition. Let N a linear operator from H to H. N is called normal if N commutes with its adjoint N ∗ . 2.6. Definition. Given a normal operator A, a complex number λ is called an eigenvalue or punctual spectral value of A if the operator A − λI is not one to one. A complex number λ is called a continuous spectral value if the operator A − λI is one to one and the operator (A − λI)−1 is densely defined but is unbounded. 2.7. Fact (Theorem 1, section 93 in [1]). A point r belongs to the continuous spectrum of an operator A if and only if there is an orthonormal sequence of elements (fn : n ∈ ω) such that limn→∞ (Afn − rfn ) = 0. 2.8. Definition. The spectrum of an operator A, denoted by σ(A), is the set of the punctual and continuous spectral values. It is well known that if A is a bounded normal operator, then σ(A) is a bounded subset of C and that if A is self-adjoint, then σ(A) is a subset of R. 2.9. Definition. A self-adjoint operator A different from the zero operator is called positive and we write A ≥ 0, if hAu|ui ≥ 0 for all u ∈ H. If A and B are selfadjoint operators, we write A ≥ B if A − B is positive. 2.10. Proposition. For a self-adjoint operator A, A2 is positive. Proof. Clear.
2.11. Definition. A self-adjoint Q operator is called a square root of a positive self-adjoint operator A if Q2 = A. 2.12. Fact (Theorem 1, section 36 in [12]). Let A be a self-adjoint operator and let E+ be the projection of H onto the null space of the operator A − Q where Q is the positive square root of A2 . Then (1) Any bounded operator R that commutes with A, commutes with E+ . (2) AE+ ≥ 0, A(Id − E+ ) ≤ 0. 2.13. Fact (Theorem 2, section 36 in [12]). Let A be a bounded self-adjoint operator, let r be a real number and let E+ (r) the projection operator constructed for Ar = A−rI according to Fact 2.12. If we denote by Er the projection operator Id−E+ (r), then the family {Er }r∈R satisfies the following conditions:
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
(1) Any bounded operator R that commutes with A commutes with Er . (2) Er ≤ Es if r < s. (3) Er is continuous on the left: ∪s 0, A = m rdEr , where the integral is to be interpreted as limit of finite sums in the sense of uniform convergence in the space of operators, and m = inf(σ(A)) and M = sup(σ(A)). Let A as above. The Spectral Decomposition Theorem can be interpreted in the following way: given > 0 and δ > 0 there is an N such that for any partition m = r0 < s0 = r1 < s1 = r2 < · · · < sN = M + with max{sk − rk } < 2(M + )/N , PN if we write E(∆k ) for Esk − Erk , we have kA − k1 rk E(∆k )k < δ. 2.16. Definition. Let A be a normal operator. The real and imaginary parts of A are the operators Ar = 21 (A + A∗ ) and Ai = 12 (A − A∗ ) 2.17. Fact. The real and imaginary parts Ar and Ai of a normal operator A are self-adjoint. Proof. For u, v ∈ H, hAr u|vi = h 12 (A + A∗ )u|vi = hu| 12 (A∗ + A∗∗ iv = hu| 21 (A + A∗ )vi = hu|Ar vi. Similarly for Ai .
2.18. Corollary. If A is a normal operator, then Ar and Ai have integral repreRb Rb sentations Ar = a sdEs and Ai = a tdFt , where Es Ft = Ft Es , and we can write RR A= (s + it)dEs dFt . 2.19. Definition. A normal operator U on H is said to be unitary if U is a bijection and for any u, v ∈ H, hU u|U vi = hu|vi. 2.20. Fact (Section 71 in [1]). Let U be a unitary operator. Then there is a unique Rπ resolution of the identity {Er |r ∈ [0, 2π]} such that U k = 0 eikt dEt for all k ∈ Z.
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We want to know if we can recover 2.21. Theorem (Theorem 2.6.3 in [3]). Let N be a normal operator on H, let B(σ(N )) the algebra of Borel functions from σ(N ) into the complex numbers, and let B(H) the algebra of linear operators on H. Then there exist a isometric monomorphism π : B(σ(N )) → B(H) such that π(f¯) = (π(f )∗ ), π(1) = Id and if f = P P P P i j ¯ , then π(f ) = i j aij N i (N ∗ )j , where by 1 we denote the constant i j aij z z function on σ(N ) with value 1. 2.22. Fact. Let A be a selfadjoint operator on H. Let m = inf{σ(A)} and M = sup(σ(A)). Let f the function on [m, M ] such that f (x) = 0 if x ≤ 0 and f (x) = 1 if x > 0. Then f is aproximable by polynomials. Proof. This function clearly belongs to L2 [m, M ] and the polynomials are dense in L2 [m.M ], so the conclusion follows.
2.23. Theorem. Let A be a selfadjoint operator. Let E+ be the projection of H onto the null space of the operator A − Q where Q is the positive square root of A2 . Then E+ is approximable by operators of the form g(A) where g is a polynomial. Proof. For any v ∈ H we have E+ (v) =
RM m
g(r)dEr (v) where g is the function
defined in Fact 2.22. But by 2.22 g(x) can approximated by polynomials, E+ can also be approximated by polynomials in A.
It is important to note that for a selfadjoint operator A on a Hilbert space H, E+ is the limit of a sequence of polynomials in A, but that the sequence does not converge uniformly. That is, given a unit vector v ∈ H, E+ (v) can be approximated by polynomials in A(v), but the rate of convergence depends on v. 2.24. Definition. The essential spectrum σe (U ) of a linear operator U is the set of accumulation points of the spectrum of U together with the set of eigenvalues of infinite multiplicity. The finite spectrum σf in (U ) of a linear operator U , the complement of σe (U ), is the set of isolated points of the spectrum of U of finite multiplicity.
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
3. Unitary operators We begin this section by characterizing the theory of the expansion of a Hilbert space with a unitary operator in terms of the spectrum of the operator using an instance of a result by C. Ward Henson: 3.1. Theorem. Let A and B be two normal operators on Hilbert spaces HA , HB respectively. Then the structures (HA , +, 0, h|i, A) and (HB , +, 0, h|i, B) are elementarily equivalent if and only if (1) σe (A) = σe (B). (2) dim{x ∈ HA : Ax = λx} = dim{x ∈ HB : Bx = λx} for λ ∈ S 1 \ σe (A). We will provide a proof for the result when the operators involved have a countable spectrum and are unitary. Henson’s original proof (unpublished), based on a Theorem by Voiculescu (see [10]), yields stronger information than our proof, in particular it shows that the theory T h(H, NA ) is separably categorical up to perturbations of the automorphism. We follow the standard way of dealing with Hilbert spaces, we write the structure as (H, +, 0, h|i) and we work inside the unit ball. We denote by L the language of Hilbert spaces and by LU the language of Hilbert spaces with a new unary function, which we denote as U . Whenever we quantify over H we mean we are quantifying over its unit ball. 3.2. Notation. For λ ∈ σ(U ), we denote by Hλ the set {x ∈ H : U x = λx}, by Pλ the projection operator onto the space Hλ . 3.3. Theorem. Let UA and UB be two unitary operators with countable spectrum on Hilbert space H. Then the structures (HA , +, 0, h|i, UA ) and (HB , +, 0, h|i, UB ) are elementarily equivalent if and only if (1) σe (UA ) = σe (UB ). (2) dim{x ∈ HA : UA x = λx} = dim{x ∈ HB : UB x = λx} for λ ∈ S 1 \σe (UA ). Proof.
⇒) We may assume the structures are separable and we will write H instead of HA , HB .
Assume that the structures (H, +, 0, h|i, UA ) and
(H, +, 0, h|i, UB ) are elementarily equivalent. Let σ A be the spectrum of UA and let σ B be the spectrum of UB . For every µ ∈ S 1 \ σ A , let η = d(µ, σ A ).
HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR
7
· kU u−µuk = 0 is true for (H, +, 0, h|i, UA ) Then the statement supu ηkuk− and thus true for (H, +, 0, h|i, UB ). Therefore σ B ⊂ σ A . In a similar way we can show that σ A ⊂ σ B . For every λ ∈ σfAin whose eigenspace is of dimension mλ , we have that the statement inf
u1 u2 ···umλ
max |hui |uj i|, |kui k − 1|, |U (ui ) − λui | = 0
is valid in (H, +, 0, h|i, UA ). Thus the same statement is true for (H, +, 0, h|i, UB ). From this condition and the fact that λ is an isolated point in the spectrum it easily follows that dim{x ∈ H : UA x = λx} ≤ dim{x ∈ H : UB x = λx}. In a similar way we can prove that dim{x ∈ H : UA x = λx} ≥ dim{x ∈ H : UB x = λx}. Also, for every λ ∈ σeA and every k ≥ 1, inf
u1 u2 ···uk
max |hui |uj i|, |kui k − 1|, |U (ui ) − λui | = 0
is true for (H, +, 0h|i, UA ) and thus also true for (H, +, 0, h|i, UB ). This implies that σeA = σeB . ⇐) Conversely, assume that σeA = σeB and dim{x ∈ H : UA x = λx} = dim{x ∈ H : UB x = λx} for λ ∈ σfAin . Let σe = σeA . We may assume, exchanging (HA , UA ) and (HB , UB ) for elementary superstructures, that (HA , UA ) and f in λ λ = {x ∈ , where HA (HB , UB ) are ℵ0 -saturated. Let HA = ⊕λ∈σf in HA λ HA : UA (x) = λx}. Note that HA is the domain of a finite dimensional f in λ λ = ⊕λ∈σf in HB Hilbert subspace of HA . Let HB , where HB = {x ∈ HB : f in UB (x) = λx}. (HB , UB ) is a substructure of (HB , UB ). Then for each f in f in λ λ λ ∈ σf in , (HA , UA ) ∼ , UB ), so (HA , UA ) ∼ = (HB = (HB , UB ). We denote
this common substructure as Hf in . Let HC = HA ⊕Hf in HB , the free amalgamation of HA and HB over Hf in and let UC be the induced unitary map on HC determined by UA and UB . We will prove that (HA , UA ) (HC , UC ) and that (HB , UB ) (HC , UC ). From this we get that the structures (HA , +, 0, h|i, A) and (HB , +, 0, h|i, B) are elementarily equivalent. Claim (HA , A) (HC , C) We use the Tarski-Vaught test. Let ϕ(x1 , . . . , xn , y) be a formula and n let a ¯ = (a1 , . . . , an ) ∈ HA . Assume that inf{ϕ(HC ,UC ) (a1 , . . . , an , b) : b ∈
8
CAMILO ARGOTY AND ALEXANDER BERENSTEIN
HC } = r and let > 0. Let b ∈ HC be such that ϕ(a1 , . . . , an , b) − r < . We may write b = bA + d, where bA = PHA (b), so d ⊥ HA . There exists k ≥ 1 and an L-formula ψ(x11 , . . . , xn1 , y1 ; . . . ; x1k , . . . , xnk , yk )
such that (HC , C) |= supx1 . . . supxn supy |ϕ(x1 , . . . , xn , y)−ψ(x1 , . . . , xn , y; . . . ; U k (x1 ), . . . , U k (xk ), U k (y 0. Clearly d ∈ ⊕λ∈σe HλB , so d =
P
λ∈σe
Pλ (d). Note that Pλ (d) ⊥ HA so
λ Pλ (d) ⊥ Pλ (ai ) for each i ≤ n. Since (HA , UA ) is ℵ0 -saturated, dim(HA )= λ ∞ for each λ ∈ σe , so there exists cλ ∈ HA such that kcλ k = kPλ (d)k and P cλ ⊥ Pλ (a1 ), . . . , Pλ (an ), Pλ (bA ). Let c = λ∈σe cλ and let b0 = bA + c.
Then tp(c, . . . , U k (c)/{¯ a, U (¯ a), . . . , U k (¯ a), bA }) = tp(d, . . . , U k (d)/{¯ a, U (¯ a), . . . , U k (¯ a), bA }), so tp(b, . . . , U k (b)/{¯ a, U (¯ a), . . . , U k (¯ a)}) = tp(b0 , . . . , U k (b0 )/{¯ a, U (¯ a), . . . , U k (¯ a)}), so ψ(a1 , . . . , an , b, . . . , U k (a1 ), . . . , U k (ak ), U k (b)) = ψ(a1 , . . . , an , b0 , . . . , U k (a1 ), . . . , U k (ak ), U k (b0 )) and ϕ(a1 , . . . , an , b0 ) − r < . Since was arbitrary we get inf{ϕ(HC ,UC ) (a1 , . . . , an , b) : b ∈ HC } = r = inf{ϕ(HC ,UC ) (a1 , . . . , an , b) : b0 ∈ HA } In other words, since Theorem ?? this theorem means that the operator theoretic concept of aproximate unitary equivalence and the model theoretic concept of elementary equivalence. The previous Theorem is proved in [6] when σ A = σeA = S 1 . Let U be a unitary operator in a separable Hilbert space H and σ be the corresponding spectrum. The previous result says that the theory of the structure (H, U ) = (H, +, 0, h, i, U ) determines the dimension of the eigenspace corresponding to the isolated points of the spectrum. We strengthen this observation by showing the definability of these eigenspaces. 3.4. Notation. For λ ∈ σ, we denote by Bλ the unit ball in the space Hλ . Recall that a closed set D ⊂ H is definable if the function dist(x, D) is a definable predicate. We will use the following characterization of definable sets and functions: 3.5. Fact (Proposition 9.18 of [4]). Let D ⊆ H be closed. Then D is definable in (H, U ) if and only if there is a definable predicate P : H n → [0, 1] such that
HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR
9
P (x) = 0 for all x ∈ D and ∀∃δ∀x ∈ H n (P (x) ≤ δ ⇒ d(x, D) ≤ ). 3.6. Fact (Proposition 9.22 of [4]). Let (H0 , U 0 ) be κ-saturated where κ is uncountn
able and let A ⊆ H 0 have cardinality < κ. Let f : H 0 → H 0 be any function and let Gf its graph. Then the following are equivalent: (1) f is definable in (H0 , U 0 ) over A. (2) Gf is type-definable in (H0 , U 0 ) over A. 3.7. Lemma. Let λ ∈ σ be isolated, let χ = d(λ, σ \ {λ}) and let u ∈ H. If kU (u) − λuk < then ku − Pλ (u)k
0∃N ∀n ≥ Pn N, kb − k=0,λ∈σ ck,λ Pλ (ak,λ )k < . Let R(x) = kx − bk and φn (x) := kx − Pn k=0,λ∈σ ck Pλ (ak,λ )k. Then ∀ > 0∃N ∀n ≥ N ∀x(|R(x) − φn (x)| < ) and {b} is definable over A.
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13
Let b 6∈ E, then for some λ0 ∈ σ, Pλ0 (b) 6∈ E. Since there are infinitely many vectors in Hλ0 ∩ E ⊥ with norm kPλ0 (b − PE (b))k, there are infinitely many realizations of tp(b/A) and therefore b 6∈ dcl(A).
3.16. Lemma. Let A ⊆ H. Then acl(A) is the closed Hilbert space generated by the union of dcl(A) with all the finite dimensional subspaces Hλ with λ ∈ σf in . Proof. Every ball Bλ for λ ∈ σf in is algebraic over ∅ because the closed unit ball of a finite dimensional space is compact. Therefore dcl(A) ∪λ∈σf in Hλ ⊂ acl(A). Conversely, let E be the closure of the space generated by dcl(A) and ∪λ∈σf in Hλ . If b 6∈ E, then Pλ0 (b) − Pλ0 (PE (b)) 6= 0 for some λ0 ∈ σe . Without loss of generality the dimension of Hλ0 ∩dcl(A)⊥ is infinite and the set {b0 ∈ H : tp(b0 /A) = tp(b/A)} is unbounded, thus b 6∈ acl(A).
3.17. Proposition. Let p, q ∈ S1 (∅) and let a |= p, b |= q. Then d(p, q) = pP 2 λ∈σ (kPλ (a)k − kPλ (b)k) . Proof.
00
≥00 . It is easy to see that: p
kak2 − 2ha|bi + kbk2 = sX = (kPλ (a)k2 − 2hPλ (a)|Pλ (b)i + kPλ (b)k2 )
ka − bk =
λ∈σ
≥
sX
(kPλ (a)k2 − 2kPλ (a)kkPλ (b)k + kPλ (b)k2 )
λ∈σ
=
sX
(kPλ (a)k − kPλ (b)k)2
λ∈σ
Then, d(p, q)
inf{ka0 − b0 k | H |= p(a0 ) and H |= q(b0 )} sX (kPλ (a)k − kPλ (b)k)2 . ≥ =
λ∈σ 00
≤00 . Let a0 , b0 be such that tp(a) = tp(a0 ) and tp(b) = tp(b0 ). Then kPλ (a)k =
kPλ (a0 )k and kPλ (b)k = kPλ (b0 )k for λ ∈ σ, but the inner products hPλ (a0 )|Pλ (b0 )i depend on the choice of a0 and b0 . We may choose a0 and b0 such that for each
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
λ ∈ σ, hPλ (a0 )|Pλ (b0 )i = kPλ (a0 )kkPλ (b0 )k. Then, d(p, q)
=
inf{ka − bk|H |= p(a) and H |= q(b)}
ka0 − b0 k = p = ka0 k2 − 2ha0 |b0 i + kb0 k2 sX (kPλ (a0 )k2 − 2hPλ (a0 )|Pλ (b0 )i + kPλ (b0 )k2 ) = = ≤
λ∈σ
=
sX
(kPλ (a0 )k2 − 2kPλ (a0 )kkPλ (b0 )k + kPλ (b0 )k2 ) =
λ∈σ
=
sX
(kPλ (a)k − kPλ (b)k)2 .
λ∈σ
3.18. Definition. Recall that the theory of a metric structure M is called separably categorical if whenever N1 , N2 |= T h(M) are separable we have N1 ∼ = N2 . 3.19. Lemma. The theory Tσ is separably categorical. Proof. Let (H0 , U 0 ) and (H00 , U 00 ) be separable models of Tσ . Then for each λ ∈ σ, dim(Hλ0 ) = dim(Hλ00 ) (which is either finite or ℵ0 ). Hence, we have that for every λ ∈ σ, Hλ0 ∼ = Hλ00 and thus (H0 , U 0 ) ∼ = (H00 , U 00 ).
One could also prove the previous Lemma using Proposition 3.17. It is clear that the formula presented in Proposition 3.17 is definable and thus the logic topology and the distance topology agree on the space of types and by Theorem 12.4 [4] Tσ is separably categorical. 3.20. Definition. The theory of a metric structure Mis called ω-stable if for any N |= T h(M) and A ⊆ N countable, S(A) is separable. 3.21. Theorem. The theory Tσ is ω-stable. Proof. Let H |= Tσ be separable and let A ⊆ H be a countable set. Let A¯ be the L algebraic closure of A and write H = A¯ + A¯⊥ = λ∈σ Hλ . For each λ ∈ σe , let Hλ0 be a separable Hilbert space such that Hλ ∩ Hλ0 = {0} and let H 0 = H ⊕λ∈σe Hλ0 . We define for each v ∈ Hλ0 , Uλ0 (v) = λv and let U 0 be the unitary map on H0 induced by U and Uλ0 , λ ∈ σe . Then (H0 , U 0 ) |= Tσ and dim(Hλ0 ∩ A¯⊥ ) = ∞ for
HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR
15
each λ ∈ σe . In particular, by Theorem 3.14 H0 realizes all types over A, so we can work inside the structure H0 . Since H0 is separable, so is S(A).
3.2. Countably infinite spectrum. Our next step is to consider a unitary operator U with countable spectrum σ. We write σ = σa ∪ σi , where σi are the isolated points of the spectrum and σa are the non-isolated points of the spectrum. 3.22. Theorem. The following properties are true: (1) Tσ has quantifier elimination. Furtheromore, for any a ¯ ∈ Hn , tp(¯ a) is determined by the values hPλ (ak ), Pλ (aj )i for λ ∈ σ, j, l ≤ n. (2) Tσ is ω-stable. Proof.
(1) It follows from Lemma ?? that for any v ∈ H and λ ∈ σ, Pλ (v) is in the quantifier free definable closure of v. The rest o the proof follows as in Theorem 3.14.
(2) We can proceed as in Theorem 3.20. 3.23. Proposition. Let p, q ∈ S1 (∅) and let a |= p, b |= q. Then d(p, q) = pP 2 λ∈σ (kPλ (a)k − kPλ (b)k) . Proof. Similar to Proposition 3.17.
3.24. Theorem. The principal types in S1 (T ) are the ones of elements a ∈ H with Pλ (a) = 0 for λ ∈ σa . Proof. We can build a structure (H0 , 0, +, h|i, U 0 ) elementarily equivalent to (H, 0, +, h|i, U ) such that Hλ = 0 for all λ ∈ σa . In this structure the types of elements a such that Pλ (a) 6= 0 for some λ ∈ σa are omitted. Conversely, assume that a is such that Pλ (a) = 0 for all λ ∈ σa and let (H0 , 0, +, h, i, U 0 ) be a model of Tσ . By Theorem 3.10 the projections Pλ are definable for every isolated λ ∈ σ and thus Hλ0 6= 0. Let a0λ ∈ Hλ0 be such that P 0 0 0 ka0λ k = kPλ (a)k for each λ ∈ σi and let a0 = λ∈σi aλ . Then a ∈ H and tp(a) = tp(a0 ). One could also use Proposition 3.23 to prove the previous Theorem.
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
3.25. Corollary. The atomic models of Tσ are the models in which the accumulation points of the spectrum are not eigenvalues. 3.26. Corollary. The ℵ0 -saturated models of Tσ are the ones in which the accumulation points of the spectrum are eigenvalues whose eigenspace is infinite dimensional. 3.27. Corollary. For λ ∈ σa , the set Hλ is not definable and the function Pλ is not definable. Proof. Assume, in order to get a contradiction, that Hλ is definable. In an ℵ0 saturated structure the statement inf u∈Hλ (kuk − 1)2 = 0 is true. Thus in the prime model this property also holds and there is a vector of norm one in Hλ , a contradiction. Assume now that Pλ is definable. Then kx − Pλ (x)k measures the distance from x into Hλ (x) and thus Hλ is definable, a contradiction.
3.28. Theorem. If σa is finite then Tσ has ℵ0 nonisomorphic separable models. If σa is infinite then Tσ has 2ℵ0 nonisomorphic separable models. L Proof. Let H and H0 be separable models of Tσ . We can write H = λ∈σi ∪σa Hλ L and H0 = λ∈σi ∪σa Hλ0 . For λ ∈ σi , Hλ ∼ = Hλ0 , so the only diference between H and H0 can come from the spaces Hλ and Hλ0 for λ ∈ σa . For such λ, the spaces Hλ and Hλ0 are isomorphic if and only if dim(Hλ ) = dim(Hλ0 ). Assume first that σa is a finite non-empty set. So there is exactly one model up to isomorphism for every possible dimension of Hλ for λ ∈ σa . Thus there are ℵ0 nonisomorphic separable models of the theory Tσ . On the other hand, if σa is an infinite countable set, there are 2ℵ0 nonisomorphic separable models of Tσ .
3.3. Forking. We fix a countable spectrum σ and a structure (H, U ) |= Tσ which is κ-saturated and strongly κ-homogeneous for some uncountable inaccessible cardinal κ. We say A ⊂ H is small if |A| < κ. 3.29. Definition.
(1) Let a ¯ = (a1 , . . . , am ) ∈ H m ; let B, C ⊆ H be small, let
∗ ¯ is ∗B ∪ C = acl(B ∪ C) and C¯ = acl(C). We write a ¯^ | C B and say that a
independent from B over C if PB∪C (Pλ (ai )) = PC¯ (Pλ (ai )) for i = 1, . . . , m and λ ∈ σ. ∗
(2) For A, B, C ⊆ H small we say A ^ | C B if and only if for all finite subsets a ¯ ∗ of A, we have that a ¯^ | C B.
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3.30. Lemma. Let B be an algebraically closed small set in (H, U ). Then for every λ ∈ σ and for every a ∈ H, Pλ (PB (a)) = PB (Pλ (a)). Proof. We first need the following claim: 3.31. Claim. PB and U commute Proof. Let a ∈ H. We can write a = aB + aB ⊥ , where aB = PB (a) and aB ⊥ = a − aB . Then U (a) = U (aB ) + U (aB ⊥ ). For any b ∈ B, U (b), U ∗ (b) ∈ B. Thus hb|U (aB ⊥ )i = hU ∗ (b)|aB ⊥ i = 0. This implies that U (aB ) ∈ B and U (aB ⊥ ) ∈ B ⊥ ; PB (U (a)) = PB (U (aB ) + U (aB ⊥ )) = U (aB ) and thus PB (U (a)) = U (PB (a)).
The result follows from Fact 2.13 part 1.
3.32. Notation. For A ⊂ H small, we write A¯ for acl(A). ∗
3.33. Corollary. Let a ¯ = (a1 , . . . , am ) ∈ H m ; let B, C ⊆ H be small. Then a ¯^ | CB if and only if PB∪C (ai ) = PC (ai ) for i ≤ m. ∗
Proof. Assume first that a ¯^ | C B, then PB∪C (Pλ (ai )) = PC¯ (Pλ (ai )) for any i ≤ n. By Lemma 3.30 we get Pλ (PB∪C (ai )) = Pλ (PC¯ (ai )) for all λ and thus PB∪C (ai ) = PC¯ (ai ). The converse is proved in a similar way.
3.34. Proposition. Given two tuples a ¯ = (a1 , . . . , an ) and ¯b = (b1 , . . . , bm ) and a ∗
∗
small set C, a ¯^ | C ¯b if and only if Pλ (ak ) ^ | C Pλ (bj ) for k = 1, . . . , n; j = 1, . . . , m and λ ∈ σ. ∗ Proof. Let a ¯ = (a1 , . . . , an ) and ¯b = (b1 , . . . , bm ). If a ¯^ | C ¯b, then we have that, P{b1 ,...,bm }∪C (Pλ (ai )) = PC¯ (Pλ (ai )) for every i = 1, . . . , n and λ ∈ σ.
For λ ∈ σ and j ≤ m, we have C¯ ⊂ C ∪ {Pλ (bj )} ⊆ C ∪ {bj } ⊆ C ∪ {b1 , . . . , bm }, so for every i = 1, . . . , n, P{Pλ (bj )}∪C (Pλ (ai )) = PC¯ (Pλ (ai )). Conversely, let us suppose that P{Pλ (bj )}∪C (Pλ (ak )) = PC¯ (Pλ (ak )) for all λ ∈ σ, k = 1, . . . , n, j = 1, . . . , m. We fix λ ∈ σ. We can write Pλ (ak ) = PC¯ (Pλ (ak ))+PC¯ ⊥ (Pλ (ak )). Since P{Pλ (bj )}∪C (Pλ (ak )) = PC¯ (Pλ (ak )) for every j = 1, . . . , m, then PC¯ ⊥ (Pλ (ak )) ⊥ Pλ (bj ) for j = 1, . . . , m and PC¯ ⊥ (Pλ (ak )) ⊥ C ∪ {Pλ (bj )} for j = 1, . . . , m.
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
Thus PC¯ ⊥ (Pλ (ak )) ⊥ C ∪ {Pλ (b1 ), . . . , Pλ (bm )} and PC∪{Pλ (b1 ),...,Pλ (bm )} (Pλ (ak )) = PC¯ (Pλ (ak )). Since PC∪{b1 ,...,bn } (Pλ (ak )) belongs to the closed space Hλ , we also get PC∪{Pλ (b1 ),...,Pλ (bm )} (Pλ (ak )) = PC∪{b1 ,...,bm } (Pλ (ak )). ∗
¯^ | C ¯b. So PC∪{b1 ,...,bm } (Pλ (ai )) = PC¯ (Pλ (ai )) for any λ ∈ σ and then a
3.35. Lemma. Let C ⊆ H be algebraically closed and let a ¯ = (a1 , . . . , an ) ∈ H n , ∗ ¯b = (b1 . . . , bm ) ∈ H m be tuples in H. Then a ¯^ | C ¯b if and only if Pλ (ak ) − PC¯ (Pλ (ak )) ⊥ Pλ (bj ) for k = 1, . . . , n, j = 1, . . . , m and λ ∈ σ.
Proof. Given two tuples a ¯ = (a1 , . . . , an ) and ¯b = (b1 , . . . , bm ), by Proposition 3.34 ∗
∗
a ¯^ | C ¯b if and only if Pλ (ak ) ^ | C Pλ (bj ) for k = 1, . . . , n, j = 1, . . . , m and λ ∈ σ. This happens if and only if PC∪{Pλ (bj )} (Pλ (ak )) = PC¯ (Pλ (ak )) for k = 1 . . . , n, j = 1, . . . , m and λ ∈ σ. Finally, PC∪{Pλ (bj )} (Pλ (ak )) = PC¯ (Pλ (ak )) if and only if Pλ (ak ) − PC¯ (Pλ (ak )) ⊥ Pλ (bj ).
∗
3.36. Theorem. The relation ^ | satisfies the following properties: finite character, local character, transitivity, symmetry, invariance, existence and stationarity. Proof. We prove all the properties: ∗
∗
(1) Finite character: we show that a ¯^ | C B if and only if a ¯^ | C B0 for all finite ∗
∗
∗
B0 ⊆ B. First of all, if a ¯^ | C B and B 0 ⊆ B then a ¯^ | C B 0 . If a ¯^ 6 | C B, PB∪C (ak ) 6= PC¯ (ak ) for some 1 ≤ k ≤ n. Let b = PB∪C (ak ) − PC¯ (ak ). Then there exist b1 , . . . , bl ∈ B, c1 , . . . , cm ∈ C, λ1 , . . . , λl+m ∈ σ and Pl Pm α1 , . . . , αn , β1 , . . . , βm ∈ C such that |b− k=1 αk Pλk (bk )− j=1 βj Pλl+j (cj )| < ∗
kbk/2. Let B0 = {b1 , . . . , bl }, then a ¯^ 6 | C B0 . (2) Local Character: For every a and B there exists a sequence (dn : n ∈ ω) such that dn → PB¯ (a) and dn = α1n Pλ1n (b1n ) + · · · + αkn n Pλkn n (bkn n ) where bij ∈ B, λij ∈ σ and αij ∈ C. Let B0 = {bij : i ≤ j, j ∈ ω}. B0 ⊆ B ∗
is countable and a ^ | B B. 0 (3) Transitivity of independence: Let A ⊆ B ⊆ C ⊂ H be small and let ∗
a ¯ = (a1 , . . . , an ). If a ¯^ | A C, PC¯ (ak ) = PA¯ (ak ) for k = 1, . . . , n. So, ∗ PC¯ (ak ) = PB¯ (ak ) = PA¯ (ak ) for k = 1, . . . , n and therefore a ¯^ | B C and ∗
a ¯^ | A B. The converse is proved in a similar way. (4) Symmetry: It enough to show that for any tuples a ¯ = (a1 , . . . , an ) and ∗ ∗ ¯b = (b1 , . . . , bm ) and small sets C, a ¯^ | C ¯b if and only if ¯b ^ | Ca ¯. Let
HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR
19
λ ∈ σ, k ≤ n and j ≤ m, we can write Pλ (ak ) = Pλ (akC¯ ) + Pλ (akC¯ ⊥ ) and Pλ (bj ) = Pλ (bjC¯ ) + Pλ (bjC¯ ⊥ ), where akC¯ = PC¯ (ak ), akC¯ ⊥ = PC¯ ⊥ (ak ). If ∗
a ¯^ | C ¯b by Lemma 3.35 Pλ (ak ) − Pλ (akC¯ ) ⊥ Pλ (bj ) then hPλ (akC¯ ⊥ )|Pλ (bjC¯ ) + Pλ (bjC¯ ⊥ )i = 0. So hPλ (akC¯ ⊥ )|Pλ (bjC¯ )i + hPλ (akC¯ ⊥ )|Pλ (bjC¯ ⊥ )i = 0. By Lemma 3.30 hPλ (akC¯ ⊥ )|Pλ (bjC¯ )i = 0, so hPλ (akC¯ ⊥ )|Pλ (bjC¯ ⊥ )i = 0. On the other hand, by Lemma 3.30 hPλ (akC¯ )|Pλ (bjC¯ ⊥ )i = 0, so hPλ (ak )|Pλ (bjC¯ ⊥ )i = 0. Thus hPλ (ak )|Pλ (bj ) − Pλ (bjC¯ )i = 0 and therefore Pλ (ak ) ⊥ Pλ (bj ) − Pλ (PC¯ (bj )). ∗
By Lemma 3.35, this implies that ¯b ^ | Ca ¯ which completes the proof (5) Invariance: For every u, v ∈ H and f ∈ Aut(H), hu|vi = hf (u)|f (v)i. ∗ Let a ¯ = (a1 , . . . , an ) and ¯b = (b1 , . . . , bm ). Then a ¯^ | C ¯b if and only if Pλ (ak ) − Pλ (PC¯ (ak )) ⊥ Pλ (bj ) for every k = 1, . . . , n, j = 1, . . . , m and
λ ∈ σ if and only if Pλ (f (ak )) − Pλ (Pf (C) ¯ (f (ak ))) ⊥ Pλ (f (bj )). (6) Existence: Let a ¯ = (a1 , . . . , an ) ∈ H n and let A ⊆ H be small. By quantifier elimination, the type tp(¯ a/A) is determined by PA¯ (Pλ (ak )) for λ ∈ σ and the inner products hPλ (ak )|Pλ (aj )i for k, j ≤ n. Let ¯b ∈ H n and B ⊇ A be small. tp(¯b/B) is a free extension of tp(¯ a/A) if and only if tp(¯b/A) = tp(¯ a/A) and PB¯ (Pλ (bk )) = PA¯ (Pλ (ak )) for all λ ∈ σ and k = 1, . . . , n. For each λ ∈ σ, let aλkA¯⊥ = PA¯⊥ (Pλ (ak )), aλkA¯ = PA¯ (Pλ (ak )). Since B is small, for each λ ∈ σe , dim(Hλ ∩ B ⊥ ) = ∞, so we can find P dλk ∈ Hλ ∩ B ⊥ with kaλkA¯⊥ k = kdλk k. Let bk = λ∈σ (aλkA¯ + dλk ) and let ¯b = (b1 , . . . , bn ). Then tp(¯ a/A) = tp(¯b/A) and PB¯ (Pλ (bk )) = PA¯ (Pλ (ak )) for k = 1, . . . n. (7) Stationarity: Let a ¯ = (a1 , . . . , an ), ¯b = (b1 , . . . , bn ), ¯b0 = (b01 , . . . , b0n ) ∈ H n and let A ⊆ B ⊆ H be small. Assume that the types of ¯b and ¯b0 over B are free extensions of tp(¯ a/A). Then tp(¯b/A) = tp(¯b0 /A) and for every λ ∈ σ and i ≤ n, PB¯ (Pλ (bi )) = PA¯ (Pλ (bi )) = PA¯ (Pλ (b0i )) = PB¯ (Pλ (b0i )). Thus tp(¯b/B) = tp(¯b0 /B). Therefore for every p ∈ S(A) |{tp(¯ a/B)|¯ a^ | A B, p ⊆ tp(¯ a/B)}| = 1.
3.37. Corollary. The theory Tσ is stable and ∗-independence coincides with the notion of independence induced by forking.
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CAMILO ARGOTY AND ALEXANDER BERENSTEIN
3.38. Observation. We had shown in Theorem 3.20 that Tσ is ω-stable and thus it has prime models over sets. Let A ⊂ H be small. For each λ ∈ σi such that ¯ < ℵ0 , let H 0 be a subspace of Hλ of dimension ℵ0 . dim(Hλ ) = ∞ and dim(Pλ (A)) λ Then A ⊕λ∈σi Hλ0 is the prime model over A of Tσ . It also follows from ω-stability that for every infinite cardinal µ, Tσ has a µsaturated model of dimension µ. Indeed, for each λ ∈ σe , let Hλ0 be a subspace of Hλ of dimension µ. Then ⊕Hλ0 |= Tσ has dimension µ and it is saturated. 3.39. Definition. Let A ⊆ H be small and p, q ∈ Sn (A). We say that p is almost orthogonal to q (p ⊥a q) if for all a ¯ |= p and ¯b |= q a ¯^ | A ¯b. 3.40. Definition. Let A ⊆ H and p, q ∈ Sn (A). We say that p is orthogonal to q (p ⊥ q) if for all B ⊇ A, pB ⊇ p a non-forking extension, and qB ⊇ q a non-forking extension, pB ⊥a qB . 3.41. Theorem. Let A ⊆ H be such that A = acl(A). Let p, q ∈ S1 (A), let a |= p and b |= q. Let a = PA (a) + a0 and b = PA (b) + b0 , let σp = {λ ∈ σe : Pλ (a0 ) 6= 0} σq = {λ ∈ σe : Pλ (b0 ) 6= 0}, then, p ⊥ q if and only if p ⊥a q if and only if σp ∩ σq = ∅. Proof. Assume that p ⊥ q, then p ⊥a q. Let a |= p and b |= q, let a0 = a − PA (a) and b0 = b − PA (b). The type tp(a/A) is determined by PA (a) and the norms of Pλ (a0 ) for λ ∈ σe . Assume, in order to get a contradiction, that Pλ0 (a0 ) 6= 0 and Pλ0 (b0 ) 6= 0 for some λ0 ∈ σe . Let a00 , b00 such that tp(a0 /A) = tp(a00 /A), tp(b0 /A) = tp(b00 /A) and Pλ0 (a00 ) is a multiple Pλ0 (b00 ). Then tp(PA (a) + a00 /A) = tp(a/A) and tp(PA (b) + b00 /A) = tp(b/A) but (PA (a) + a00 ) ^ 6 | A (PA (b) + b00 ), a contradiction to p ⊥w q. Conversely, assume that σp ∩ σq = ∅. Let B ⊇ A and let pB ⊃ p, qB ⊇ q be nonforking extensions. Let c and d realizations of pB and qB respectively. We may write c = cB¯ +c0 , d = dB¯ +d0 where cB¯ dB¯ are the projections of c and d over acl(B) = B. P P Then σpB = σp and σqB = σq . Then hc0 |d0 i = ( λ∈σp Pλ (a0 )| µ∈σq Pµ (b0 )i = P P 0 0 0 0 | B d. λ∈σp µ∈σq hPλ (c )|Pµ (d )i = 0. Then c ⊥ d and c ^ A generalization of the previous result appears in [6] when σf in = ∅ and A = ∅.
HILBERT SPACES EXPANDED WITH A UNITARY OPERATOR
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References [1] N.I. Akhiezer, I.M. Glazman, Theory of linear operators in Hilbert Space vols.I and II. Pitman Advanced Publishing Program, 1981. [2] Ita¨ı Ben Yaacov, On perturbations of continuous structures, submitted. [3] W. Arveson, A short course in spectral theory. Springer Verlag, 2002. [4] Ita¨ı Ben Yaacov, Alexander Berenstein, C. Ward Henson and Alexander Usvyatsov, Model theory for metric structures, to appear in the Proceedings of the Isaac Newton Institute’s semester on Model Theory and its applications. [5] Ita¨ı Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability. submitted. [6] Ita¨ı Ben Yaacov, Alexander Usvyatsov and Moshe Zadka, Generic automorphism of a Hilbert space, preprint. [7] Alexander Berenstein, and Steven Buechler, Simple stable homogeneous expansions of Hilbert spaces. Annals of of pure and Applied Logic. Vol. 128 (2004) pag 75-101. [8] Steven Buechler, Essential stability theory. Springer Verlag, 1991. [9] R. Dautray, L. Lions, Mathematical analysis and numerical methods for science and technology, volume 3. [10] Kenneth Davidson, C ∗ -Algebras by example, Field Institute Monographs, 1996. [11] Jos´ e Iovino, Stable theories in functional analysis University of Illinois Ph.D. Thesis, 1994. [12] L. Liusternik and V. Sobolev, Elements of Functional Analysis. Frederic Ungar Publishing Co., New York, 1961. [13] M. Reed, B. Simon, Methods of modern mathematicalphysics volume I:Functional analysis, revised and enlarged edition. Academic Press, 1980. [14] Werner Schmeidler, Linear Operators in Hilbert Space, Academic Press, 1965.
´ticas, Cra 1# Camilo Argoty, Universidad de los Andes, Departamento de Matema ´, Colombia. and, Universidad Sergio Arboleda, Departamento de Matema ´ticas, 18A-10, Bogota E-mail address: [email protected] ´ticas, Alexander Berenstein, Universidad de los Andes, Departamento de Matema ´, Colombia. Carrera 1 N 18A-10, Bogota E-mail address: [email protected]
