right spectrum and trace formula of subnormal ... - Semantic Scholar
RIGHT SPECTRUM AND TRACE FORMULA OF SUBNORMAL TUPLE OF OPERATORS OF FINITE TYPE Daoxing Xia Abstract. This paper studies pure subnormal k-tuples of operators S = (S1 , · · · , Sk ) with finite rank of self-commutators. It determines the substantial part of the conjugate of the joint point spectrum of S∗ = (S1∗ , · · · , Sk∗ ) which is the union of domains in Riemann surfaces in some algebraic varieties in Ck . The concrete form of the principal current [CP1] related to S is also determined. Besides, some operator identities are found for S. Mathematics Subject Classification (2000). Primary 47B20.
1. Introduction In this paper H is an infinite dimensional separable Hilbert space. A k-tuple of operators S = (S1 , · · · , Sk ) on H is said to be subnormal (or jointly subnormal) (cf. [Co2], [Cu2], [Pu], [EP]) if there is a k-tuple of commuting normal operators N = (N1 , · · · , Nk ) on a Hilbert space K, containing H as a subspace, satisfying Sj = Nj |H ,
j = 1, 2, · · · , k.
Without loss of generality, we may only consider the m.n.e.(minimal normal extension), i. e. there is no improper reducing subspace of N in K H. S is said to be pure, if there is no improper reducing subspace of S in H. Yakubovich [Y1], [Y2] called a subnormal operator with finite rank self-commutator as a subnormal operator of finite type, cf. also [X5], [X6], [X8]. Let us call a subnormal k-tuple of operators S = (S1 , · · · , Sk ) is of finite type if rank[Sj∗ , Sj ] < +∞, j = 1, 2, · · · , k. In this case, it is also that rank[Si∗ , Sj ] < +∞, i 6= j, i, j = 1, 2, · · · , k. In this paper we only study pure subnormal k-tuple of operators S of finite type on H.
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Daoxing Xia
The main mathematical tool of this study is the analytic model of S in [X1], [X2], [X3], [X5]. Let M = ∨kj=1 [Sj∗ , Sj ]H. (1) Then dim M < +∞, since we only consider the operators of finite type. Let def Cij = [Si∗ , Sj ]|M as an operator on the finite dimensional space M, since M reduces
[Si∗ , Sj ].
def
Λj = (Sj∗ |M )∗ ∈ L(M ),
(2) Let (3)
since M is invariant with respect to Sj∗ , j = 1, 2, · · · , k (cf.[X1]). Let def
Rij (z) = Cij (z − Λj )−1 + Λ∗i ,
(4)
and def
Qij (w, z) = (w − Λ∗i )(z − Λj ) − Cij def
Pij (z, w) = det Qij (z, w).
(5) (6)
Then Pij (z, w) = Pji (w, z). The polynomial Pij (z, w) is with leading term z ν wν where ν = dim M . For A = (A1 , · · · , Ak ), let spjp (A∗ )∗ be the conjugate of the joint point spectrum, i. e. the set of all (w1 , · · · , wk ) ∈ Ck satisfying the condition that there is a vector f ∈ H, f 6= 0 such that A∗j f = wj f, j = 1, 2, · · · , k. In §3, some relation between the right spectrum of a pure subnormal tuple S of operators of finite type and spjp (S∗ )∗ is given. In §5, we introduce a union of domains S in some Riemann surfaces which is in a domain in an algebraic variety Sa . Those S and Sa are determined basically by {Pij (·, ·)} in (6). The aim of this paper is to determine a substantial part S of spjp (S∗ )∗ for pure subnormal k-tuple of operators S of finite type in Theorem 6.1. The principal current has been studied by Pincus and Carey [PC1], [PC2], [Pi]. In this paper, we will give the concrete form Theorem 7.1 of the principal current for the pure subnormal operators of finite type. Besides, in §7, we will give some operator ∗ U+ − I = 0 for the identities for S, which are the generalization of the identity U+ unilateral shift U+ .
2. Preliminaries In order to make the proofs in this paper readable, we have to list the some basic facts in the theory of the analytic model of subnormal operators of finite type which is a special case in [X1], [X2], [X3].
Subnormal tuple of operators
3
Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators of finite type on H with m.n.e. N. Let E(·) be the spectral measure of N on sp(N). Let PM be the projection from H to M, where M is defined in (1). Let def
e(·) = PM E(·)|M
(7)
ˆ be the Hilbert space complebe the L(M )-valued positive measure on sp(N). Let H Qk −1 tion of the span of all functions j=1 (λj − uj ) a for a ∈ M and λj ∈ ρ(Sj ), j = 1, 2, · · · , k, with respect to Z (f, h) = (e(du)f (u), h(u)). sp(N)
ˆ such that Then there is a unitary operator U from H onto H (U Sj U −1 f )(u) = uj f (u),
u = (u1 , · · · , uk ) ∈ sp(N)
and (U Sj∗ U −1 f )(u) = uj (f (u) − f (Λ)) + Λ∗j f (Λ)
(8)
where f (Λ) = sp(N) e(du)f (u). This (U S1 U −1 , · · · , U Sk U −1 ) is the analytic model of S. ˆ and U = I, i. e. we Without loss of generality, we may assume that H = H only have to consider the analytic model of S. For u ∈ σ(N), (9) Qij (u, u)e(du) = 0. R
Let def
P (S) = {u ∈ Ck : Pij (u, u) = 0, i, j = 1, 2, · · · , k} Then from (8) (cf. also [X3], [PX]), sp(N) ⊂ P (S). Let def
Z
µj (z) =
sp(N)
uj − Λj e(du), z ∈ ρ(Nj ). uj − z
Then µj (·) is analytic on ρ(Nj ) and µj (z) = 0 for z ∈ ρ(Sj ). Let T = (T1 , · · · , Tk ) be a k-tuple of commuting operators on H. If there is a finite dimensional subspace K ⊂ H such that [Ti∗ , Tj ]H ⊂ K, Ti∗ K ⊂ K for i, j = 1, 2, · · · , k. Then T is said to be of finite type. In this case let def
Cij = [Ti∗ , Tj ]|K ,
def
Λi = (Ti∗ |K )∗
for i, j = 1, 2, · · · , k. Define Rij (z) = Cij (z − Λj )−1 + Λ∗i and Qij (w, z) = (w − Λ∗i )(z − Λj ) − Cij as in §1. The following lemma is useful for §4 and the future study.
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Daoxing Xia
Lemma 2.1
Let T be a k-tuple of operators of finite type. Then [Rmj (z), Rnj (z)] = 0, m, n, j = 1, 2, · · · , k,
(10)
for z ∈ ρ(Λj ). Proof Without loss of generality, we may assume that m = j = 1, n = 2. Define Ai = wi − Ti∗ , i = 1, 2, B1 = z − Ti for z ∈ ρ(T1 ), wi ∈ ρ(Ti ). Then [Ai , B1 ] = [Ti∗ , T1 ] = Ci1 PK , −1 −1 −1 −1 −1 where PK is the projection from H to K. Thus [A−1 i , B1 ] = Ai B1 Ci1 PK B1 Ai . Therefore −1 −1 −1 −1 −1 −1 −1 −1 PK A−1 1 A2 B1 |K = PK A1 [A2 , B1 ]|K + PK A1 B1 A2 |K .
(11)
The right-hand side of (11) equals −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 PK A−1 1 A2 B1 C21 PK B1 A2 |K + PK [A1 , B1 ]A2 |K + PK B1 A1 A2 |K −1 −1 −1 −1 −1 −1 −1 = PK A−1 (w2 − Λ∗2 )−1 + PK A−1 1 B1 C11 PK B1 A1 A2 |K 1 A2 B1 C21 (z1 − Λ1 )
+(z1 − Λ1 )−1 (w1 − Λ∗1 )−1 (w2 − Λ∗2 )−1 . since
A−1 i |K
= (wi − Λ∗i )−1 and
PK B1−1 |K
(12) = (z − Λ1 )−1 . On the other hand
−1 −1 −1 −1 −1 −1 −1 PK A−1 1 B1 |K = PK A1 B1 C11 PK B1 A1 |K + PK B1 A1 |K .
Thus −1 −1 PK A−1 (w1 − Λ∗1 )−1 ) = (z − Λ1 )−1 (w1 − Λ∗1 )−1 . 1 B1 |K (I − C11 (z − Λ1 )
Therefore −1 −1 . PK A−1 1 B1 |K = Q11 (w 1 , z)
(13)
From (11), (12) and (13), we have −1 −1 PK (A1 A−1 (w2 − Λ∗2 )−1 2 B1 )|K Q21 (w 2 , z)(z − Λ1 )
= (Q11 (w1 , z)−1 C11 + I)(z − Λ1 )−1 (w1 − Λ∗1 )−1 (w2 − Λ∗2 )−1 = Q11 (w1 , z)−1 (w2 − Λ∗2 )−1 Thus −1 −1 −1 = Q11 (w1 , z)−1 . PK (A−1 1 A2 B1 )|K Q21 (w 2 , z)(z − Λ1 )
Hence −1 −1 −1 PK (A−1 (w1 − R11 (z))−1 (w2 − R21 (z))−1 . 1 A2 B1 )|K = (z − Λ1 )
From [A1 , A2 ] = 0, we have [(w1 − R11 (z))−1 , (w2 − R21 (z))−1 ] = 0 which proves the lemma.
Subnormal tuple of operators
5
3. Right spectrum For a k-tuple of operators A = (A1 , · · · , Ak ) on Hilbert space H, let spr (A) = {(λ1 , · · · , λk ) ∈ Ck :
k X
(Aj − λj )(Aj − λj )∗ is not invertible }
j=1
be the right spectrum of A. It is obvious that spr (A) ⊃ spjp (A∗ )∗ . Proposition 3.1 Let S = (S1 , · · · , Sk ) be a subnormal k-tuple of operators of finite type on H, with m.n.e. N = (N1 , · · · , Nk ). If λ = (λ1 , · · · , λk ) satisfies λj ∈ σ(Sj ) ∩ ρ(Nj ), j = 1, 2, · · · , k and λ ∈ spr (S), then λ ∈ spjp (S∗ )∗ . Proof If (λ1 , · · · , λk ) ∈ spr (S), then there is a sequence {fm } ⊂ H satisfying kfm k = 1, such that k X
lim
m→∞
since
Pk
j=1 (Sj
(Sj − λj )(Sj − λj )∗ fm = 0,
j=1
− λj )(Sj − λj )∗ is self-adjoint. Thus
lim k(Sj − λj )∗ fm k = 0, j = 1, 2, · · · , k. P since k(Sj − λj )∗ fm k2 ≤ ( j (Sj − λj )(Sj − λj )∗ fm , fm ). On the other hand from (8) (cf. [X1], [X3]) m→∞
(Sj∗ − λj )fm (u) = (uj − λj )fm (u) − (uj − Λ∗j )fm (Λ), where
Z fm (Λ) =
e(du)fm (u). sp(N)
We may choose a subsequence {fmn } of {fm } such that fmn (Λ) → a ∈ M, as mn → ∞, since kfm (Λ)k ≤ 1. Therefore, from fmn (u) =
uj − Λ∗j u j − λj
fmn (Λ) +
1 (Sj∗ − λj )fmn uj − λj
and |uj − λj | ≥ dist(λj , σ(Nj )) > 0, for λj ∈ ρ(Nj ), it follows that as a sequence of vectors in H, {fmn } converges to g(u) =
uj − Λ∗j u j − λj
a, j = 1, 2, · · · , k.
On the other hand, we have fmn (Λ) = µj (λj )∗ fmn (Λ) +
Z
e(du) (Sj∗ − λj )fmn (u). uj − λj
(14)
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Daoxing Xia
Thus a = µj (λj )∗ a, j = 1, 2, · · · , k. But kgk = lim kfmk k = 1, therefore g is an eigenvector of Sj∗ corresponding to λj , for j = 1, 2, · · · , k, i. e. λ ∈ spjp (S∗ )∗ .
4. Joint eigenvectors In order to study the joint eigenvectors for Sj∗ and Sk∗ , we have to establish the following lemma. Lemma 4.1 Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators on H with m.n.e. N = (N1 , · · · , Nk ). Let zl ∈ ρ(Nl ), l = 1, 2, · · · , n, n ≤ k and w ∈ ρ(Nj ), then Z n Y e(du)(uj − Λ∗j ) = (Rνl j (w)∗ − zl )−1 µj (w)∗ + n Y (uνl − zl )(uj − w) l=1 l=1 n Z X m=1
e(du) m Y (uνl − zl )
(I + (zm − Λνm )(Rνm j (w)∗ − zm )−1 )
n Y
(Rνp j (w)∗ − zp )−1 .
p=m+1
l=1
(15) Proof For the simplicity of notation, we assume that νl = l. Let us prove (15) by mathematical induction. First, let us prove it for n = 1. From (7) and Z e(du) = (I − µj (w)∗ )Qji (w, z)−1 − Qji (w, z)−1 µi (z). (ui − z)(uj − w) (cf. [X1], [X3]). In [X1], it proves this identity in the case of i = j, but the method in the proof also applies to the case i 6= j. Therefore, Z e(du)(w − Λ∗j ) = (I − µj (w)∗ )(z − R1j (w)∗ )−1 (u1 − z)(uj − w) − (z − Λ1 )−1 (w − Rj1 (z))−1 µ1 (z)(w − Λ∗j ), since (w − Λ∗j )(z − Rij (w)∗ ) = Qji (w, z), and (w − Rji (z))(z − Λi ) = Qji (w, z). But µj (z)Rij (z) = Rij (z)µj (z) (cf. [X1], [X3]). Again, in [X1] it only proves the case of i = j. But the proof may extend to the case of i 6= j as well. We have Z e(du)(w − Λ∗j ) (u1 − z)(uj − w) = (z − R1j (w)∗ )−1 (I − µj (w)∗ ) − (z − Λ1 )−1 µ1 (z)(z − Λ1 )(z − R1j (w)∗ )−1 . From −1
(z − Λ1 )
−1
µ1 (z) = (z − Λ1 )
Z +
e(du) u1 − z
Subnormal tuple of operators
7
we have (15) for n = 1. Suppose that (15) holds for n. Then from Lemma 2.1, Z Z Z e(du)(uj − Λ∗j ) e(du) e(du) = + (w − Λ∗j ) n+1 n+1 n+1 Y Y Y (ul − zl )(uj − w) (ul − zl ) (ul − zl )(uj − w) l=1
l=1
l=1
and Z
e(du)Qj(n+1) (w, zn+1 ) n+1 Y (ul − zl )(uj − w)
Z
e(du)(−(uj − Λ∗j )(un+1 − zn+1 ) − (uj − w)(zn+1 − Λn+1 ))
l=1
=
n+1 Y
(ul − zl )(uj − w)
l=1
Z =−
e(du)(uj − Λ∗j ) − n Y (ul − zl )(uj − w) l=1
it follows that Z
Z
e(du)(zn+1 − Λn+1 ) , n+1 Y (ul − zl ) l=1
e(du)(uj − Λ∗j ) n+1 Y
(ul − zl )(uj − w)
l=1
Z =
e(du) n+1 Y
(I − (zn+1 − Λn+1 )(zn+1 − R(n+1)j (w)∗ )−1 )
(ul − zl )
l=1
Z −
e(du)(uj − Λ∗j ) (zn+1 − R(n+1)j (w)∗ )−1 . n Y (ul − zl )(uj − w) l=1
which proves the lemma. Lemma 4.2 Under condition of Lemma 4.1, let wj ∈ σ(Sj ) ∩ ρ(Nj ), j = 1, 2. Suppose there are cν ∈ C, ν = 1, · · · , k and a vector a ∈ M satisfying µj (wj )∗ a = a, j = 1, 2
(16)
Rνj (wj )∗ a = cν a, j = 1, 2; ν = 1, 2, · · · , n.
(17)
and Let def
fj (u) =
uj − Λ∗j a, j = 1, 2 uj − w j
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Daoxing Xia
Then fj (·) = f2 (·) as vectors in H and Sj∗ fj = wj fj , j = 1, 2. Proof Z
(18)
From (15), (16), (17) and Lemma 4.1, it is easy to calculate that Z e(du) e(du) f (u) = f2 (u), zl ∈ ρ(Nl ), l = 1, · · · , k. 1 k k Y Y (ul − zl ) (ul − zl ) l=1
l=1
e(du) ˆ of σ(N). Then by the Plemelj formula Let g(u) = on the continuous part L du1 X g(u)f2 (u) X g(u)f1 (u) = n n Y Y (ul − zl ) (ul − zl ) def
l=2
l=2
P for almost all points u1 in L1 = ∂σ(S1 ), where is ranging over a finite set of all points u in σ(N) with the same u1 . However zl is an arbitrary point in ρ(Nl ), l = 2, · · · , n. Therefore g(u)f1 (u) = g(u)f2 (u), for almost all u1 ∈ L1 . ˆ Similarly e({u})f1 (u) = e({u})f2 (u) Thus e(du)f1 (u) = e(du)f2 (u) for almost u ∈ L. for every point spectrum u in sp(N). Thus f1 (·) = f2 (·) as vectors in H. The formula (18) follows from (8) and (16).
5. Riemann surfaces and algebraic varieties associated with subnormal tuple of operators Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators of finite type with m.n.e. N = (N1 , · · · , Nk ). Let Oj ⊂ σ(Sj ) ∩ ρ(Nj ), j = 1, 2, · · · , k be non-empty open sets. Let Sˆj (·) be a univalent analytic function on Oj , j = 1, 2, · · · , k. Let O(O1 , Sˆ1 , · · · , Ok , Sˆk ) be defined as {z = (z1 , · · · , zk ) ∈ Ck : Pmn (Sˆm (zm ), zn ) = 0 for zj ∈ Oj , j = 1, · · · , k}. (19) Let S0 be the union of these neighborhoods {O(O1 , Sˆ1 , · · · , Ok , Sˆk )}. On each O(O1 , Sˆ1 , · · · , Ok , Sˆk ), every function f (zj ), considered as a function on z = (z1 , · · · , zj , · · · , zk ) is a local coordinate, where f is an analytic univalent function on Oj . Then S0 is a union of domains in some Riemann surfaces. Proposition 5.1 The continuum part of sp(N) ⊂ ∂S0 , the boundary of S0 . Proof Let γ be a small arc in σ(N). Let γj = {zj : (z1 , · · · , zj , · · · , zk ) ∈ γ} ⊂ σ(Nj ). Then there are domains Oj ⊂ σ(Sj )∩ρ(Nj ) and Dj ⊂ ρ(Sj ) such that Dj ∪ γj ∪Oj is a domain and γj ⊂ ∂Oj ∩∂Dj . From (9), Pij (z i , zj ) = 0, for (z1 , · · · , zk ) ∈ γ. There are analytic functions Sˆj (·) on Dj ∪ γj ∪ Oj satisfying Sj (z) = z, z ∈ γj . Actually these Sˆj (·) are branches of multivalued Schwarz functions associated with
Subnormal tuple of operators
9
those subnormal operators Sj (cf. [AS], [X5], [X6], [X7]), j = 1, 2, · · · , k. Thus Pji (Sˆj (zj ), zi ) = 0, (z1 , · · · , zk ) ∈ γ. Therefore γ ⊂ ∂O(O1 , Sˆ1 , · · · , Ok , Sˆk ). Pk Pk Suppose P (w) = j=0 pk−j wj and Q = j=0 qk−j wj are polynomials with constant coefficients pj and qi . Assume that p0 6= 0, q0 6= 0. Let def
R(P, Q) = det(rij )i,j=1,2,··· ,2k be the resolvent of P and Q, where rij = pj−i rij = qi−k−j
for 1 ≤ i ≤ j ≤ k, for 1 ≤ i − k ≤ j, k + 1 ≤ i ≤ 2k,
and rij = 0 for other pairs of i and j. It is well-known that if there is a common solution of P (·) = 0 and Q(·) = 0, then R(P, Q) = 0. Consider the algebraic variety def
Sa = ∩i6=j {(u1 , · · · , uk ) ∈ Ck : R(Pii (·, ui ), Pij (·, uj )) = 0}. It is easy to see the following. Lemma 5.1 S0 ⊂ Sa . Let γ be a small arc in the sp(N) satisfying the condition that for each j the mapping γ → γj = {uj ∈ C : (u1 , · · · , uj , · · · , uk ) ∈ γ} is one to one and γj is a simple arc (there is no node in γj ). For each γj there is a simply connected domain Oj ⊂ σ(Sj ) ∩ ρ(Nj ) such that γj is in the boundary of Oj and there is an analytic function Sˆj (·) on Oj such that Pjj (Sˆj (u), u) = 0, u ∈ Oj and the boundary value of Sˆj (·) satisfying Sˆj (z) = z, z ∈ γj , since Pjj (z, z) = 0, z ∈ γj . Let Oγ = O(O1 , Sˆ1 , · · · , Ok , Sˆk )
(20)
defined in (19) by these {Oj , Sˆj (·)}. Let S be the union of those component in S0 which contains some Oγ as a subset. This Sˆj (·) in (20) is a branch of the Schwarz function related to the subnormal operator Sj (cf. [X5], [X6], [X7], [Y1] and [Y2]). Thus the boundary of S must be also in sp(N). Therefore ∂S = sp(N). Theorem 5.1 Let S be a pure subnormal k-tuple of operators of finite type, then S ⊂ Sa . Let f be an analytic function on a neighborhood of σ(S1 )×· · ·×σ(Sk ). Define Z uj − Λj µj (z; f ) = e(du)f (u), z ∈ ρ(Nj ) σ(N) uj − z as in [X2] and [X3]. Then µj (·; f ) is an analytic function on ρ(Nj ) and µj (z; f ) = 0,
z ∈ ρ(Sj ).
10
Daoxing Xia
By Plemelj formula, the boundary value of µj (·; f ) at the continuum part Lj ⊂ σ(Nj ) is X e(du) (uj − Λj ) f (u), duj P where is ranging over a finite set of points of u ∈ sp(N) with the same j-th coordinate uj . Therefore, there are L(M )-valued analytic functions µ ˆj (u) on S0 such that on the continuum part L of sp(N) µ ˆj (u) = (uj − Λj )
e(du) . duj
It is easy to see that (uj − Λj )−1 µ ˆj (u) = (uk − Λk )−1 µ ˆk (u)
duj , for u ∈ S. duk
(21)
Remark Let π be a linear function a1 u1 +· · ·+ak uk and Sπ = a1 S1 +· · · , +ak Sk (cf. [PZ]). We may choose k linear functions π1 , π2 , · · · , πk which are linearly independent on Ck such that if we replace S1 , · · · , Sk by Sπ1 , · · · , Sπk respectively then µ ˆj (u) = µj (uj ) for u ∈ S0 .
6. S and spjp (S∗ )∗ Let Mu be the range of µ ˆj (u)∗ for u ∈ S, which is well-defined by (21) and Mu 6= {0} except a finite set F ⊂ S. Theorem 6.1 Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators of finite type. Then there is a finite set F such that S \ F ⊂ spjp (S∗ )∗ .
(22)
For w ∈ S \ F , and a ∈ Mw \ {0}, the vector f (u) =
uj − Λ∗j a, j = 1, 2, · · · , k uj − w j
(23)
satisfies f 6= 0 and Sj∗ f = wj f, j = 1, 2, · · · , k. Proof
For u in the continuum part of sp(N), X e(du) µj (u)∗ = (uj − Λ∗j ). duj
(24)
(25)
By (7) and (25), we may prove that µ ˆj (u)∗ M ⊂ µj (u)∗ M for a.e. u ∈ σ(N). Thus (I − µj (uj )∗ )ˆ µj (u)∗ = 0 for a. e. u ∈ sp(N). Therefore (I − µj (uj )∗ )ˆ µj (u)∗ = 0 for u ∈ S, since µj (·) and µ ˆj (·) are analytic functions. Hence for a ∈ Mu , w ∈ S µj (wj )∗ a = a, for j = 1, · · · , k.
Subnormal tuple of operators
11
On the other hand, from (9), we have (Sν (wν ) − Rνj (wj )∗ )ˆ µm (w)∗ = 0, w ∈ γ
(26)
for w ∈ γ and j, ν, m = 1, 2, · · · , k where {Sj (·)} are those functions in the proof of Proposition 5.1, since Sν (wν ) = wν for w ∈ γ. Thus (26) holds good for w ∈ Oγ . By analytic continuation, (26) holds good for w ∈ S. Therefore (16) and (17) are satisfied if a ∈ Mw and cν = Sν (wν ). From Lemma 4.2, it follows (23) and (24) which proves (22) and hence Theorem 6.1. Conjecture : spjp (S∗ )∗ \ S is a finite set.
7. Principal current. The short introduction of the principal current related to the present work can be seen in the introduction of [PX] or [PZ] and the papers [CP1], [CP2], [Pi]. Let us first list the following two lemmas. For a compact set σ ⊂ Cn , let A(σ) be the algebra generated by analytic functions f on a neighborhood of σ and its conjugate. Lemma 7.1 [PX] Let S be a pure subnormal k-tuple of operators of finite type with m.n.e. N. Then Z 1 mf dh trace[f (S), h(S)] = 2πi L for f, h ∈ A(sp(N)), where L is the union of a finite collection of closed curves and is also the union of a finite collection of algebraic arcs such that sp(N) is the union of L and a finite set. Furthermore, m(u) is an integer valued multiplicity function which is constant on the irreducible pieces (simple closed curves) of L. In this lemma, e(du) m(u) = 2πi · trace((uj − Λj ) ), for u ∈ L. duj Thus we have the following Lemma 7.2 The function m(·) defined in Lemma 7.1 is the boundary value of the def
function m(u) = dim Mu = rank µ ˆj (u), u ∈ S. Lemma 7.3 [PZ] Suppose that S is a pure subnormal k-tuple of finite tuple with m.n.e. N. The principal current of S l(f dh) = i · trace[f (S), h(S)] can be represented as Z 1 X l(f dh) = ml f dh 2π Cl l
where {Cl } is the collection of cycles spess (S) and the weights ml are spectral multiplicity of N at any regular point ζ of Cl .
12
Daoxing Xia
In this lemma, ml is the value of function m(u) for u ∈ Cl . By the fact that sp(N) is the boundary of S, Cartan’s formula Z Z dω = ω, σ
∂σ
Lemma 7.1-7.3 and the fact that the non-negative integer valued function m(·) is piece-wise constant. We have the following Theorem 7.1 Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators of finite type. Then for f, h ∈ A(σ(S1 ) × · · · × σ(Sk )), Z 1 mdf ∧ dh, i · trace[f (S), h(S)] = 2π S where m(·) = rank µ ˆj (·). Thus m(·) can be considered as an extension of the concept of Pincus principal function from subnormal operators to subnormal k-tuple of operators.
8. Operator identities. It is well-known that the unilateral shift U+ satisfies the identity ∗ U+ U+ − I = 0.
(27)
This identity also characterizes the unilateral shift. In this section we will introduce some operator identities for the subnormal tuple of operators of finite type which are the generalization of (27). P Let p(w, z) = pmn wm z n be a polynomial and A and B be operators. We adopt the Weyl ordering X def p(A, B) = pmn Am B n . Theorem 8.1 Let S = (S1 , · · · , Sk ) be a subnormal k-tuple of operators of finite type. Let Pij (ui , uj ) be the polynomial defined in (6) and R(Pii (·, ui ), Pij (·, uj )) be the polynomial of ui , uj defined in §5. Then for i, j = 1, 2, · · · , k, Pij (Si∗ , Sj ) = 0,
(28)
R(Pii (·, Si ), Pij (·, Sj )) = 0.
(29)
and
Proof
Suppose Pij (w, z) =
X m,n
pmn wm z n .
Subnormal tuple of operators
13
Then by (9), Pij (ui , uj ) = 0 for (u1 , · · · , uk ) ∈ sp(N). Thus for f, g ∈ H X (Pij (Si∗ , Sj )f, g) = pmn (Si∗ m Sj n f, g) X = pmn (Sj n f, Si m g) Z X = pmn (e(du)uj n f (u), ui m g(u)) sp(N) Z = (e(du)Pij (ui , uj )f (u), g(u)) = 0, sp(N)
which proves (28). Similarly, we may prove (29), since R(Pii (·, ui ), Pij (·, uj )) = 0 for (u1 , · · · , uk ) ∈ sp(N).
References [AS] D. Aharonov and H. S. Sapiro, Domains on which analytic functions satisfy quadrature identities, J. Anal. Math. 30 (1976), 39–73. [Co1] J. B. Conway, Theory of Subnormal Operators, Math. Surv. Mon. 36 (1991). [Co2] J. B. Conway, Towards a functional calculus for subnormal tuples: the minimal normal extension, 138 (1977), 543–577. [CP1] R. W. Carey and J. D. Pincus, Principal currents, Integr. Equ. Oper. Theory 8 (1985), 614–640. [CP2] R. W. Carey and J. D. Pincus, Reciprocity for Fredholm operators, Integr. Equ. Oper. Theory 90 (1986), 469–501. [Cu1] R. E. Curto, Connections between Harte and Taylor spectra, Rev. Roum. Math. Pure Appl. 31 (1986), 1–31. [Cu2] R. E. Curto, Joint hyponormality: a bridge between hyponormality and subnormality, Pro. Symp. Pure Math. 5 (1990), 69–91. [EP] J. Eschmeier and M. Putinar, Some remarks on spherical isometries, in vol. ”Systems, Approximation, Singular Integral Operators and Related Topics”(A. A. Borichev and N. K. Nikolskii, eds), Birkhauser , Bassel et al., 2001, 271-292. [Pi] J. D. Pincus, The principal index, Proc. Symp. Pure Math. 51 (1990), 373–393. [Pu] M. Putinar, Spectral inclusion for subnormal n-tuples, proc. Amer. Math. Soc. 90 (1984), 405–406.
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[PX] J. D. Pincus, D. Xia, A trace formula for subnormal tuples, Integr. Equ. Oper. Theory. 14 (1991), 469–501. [PZ] J. D. Pincus, D. Zheng, A remark on the spectral multiplicity of normal extensions of commuting subnormal operator tuples, Integr. Equ. Oper. Theory 16 (1993), 145–153. [T1] J. L. Taylor, A joint spectrum for several commuting operators, Jour. Functional Analysis 6 (1970), 172–191. [T2] J. L. Taylor, The analytic functional calculus for several commuting operators, Acta Mathematica 125 (1970), 1–38. [X1] D. Xia, On the analytic model of a subnormal operator, Integr. Equ. Oper. Theory 10 (1987), 255–289. [X2] D. Xia, Analytic theory of subnormal operators, Integr. Equ. Oper. Theory 10 (1987), 890-903. [X3] D. Xia, Analytic theory of a subnormal n-tuple of operators, Proc. Symp. Pure Math. 51 (1990), 617–640. [X4] D. Xia, Trace formulas for a class of subnormal tuple of operators, Integr. Equ. Oper. Theory 17 (1993), 417–439. [X5] D. Xia, On pure subnormal operators with finite rank self-commutators and related operator tuples, Integr. Equ. Oper. Theory 24 (1996), 107–125. [X6] D. Xia, On a class of operators with finite rank self-commutators, Integr. Equa. Oper. Theory 33 (1999), 489–506. [X7] D. Xia, Trace formulas for some operators related to quadrature domains in Riemann surfaces, Integr. Equ. Oper. Theory 47 (2003), 123–130. [X8] D. Xia, On a class of hyponormal operators of finite type, to appear in Integr. Equ. Oper. Theory. [Y1] D. V. Yakubovich, Subnormal operators of finite type I, Xia’s model and real algebraic curves, Revista Matem. Iber. 14 (1998), 95–115. [Y2] D. V. Yakubovich, Subnormal operators of finite type II, Structure theorems, Revista Matem. Iber. 14 (1998), 623–689.
Daoxing Xia Department of Mathematics Vanderbilt University Nashville, TN 37240 USA e-mail: [email protected]
1. Introduction In this paper H is an infinite dimensional separable Hilbert space. A k-tuple of operators S = (S1 , · · · , Sk ) on H is said to be subnormal (or jointly subnormal) (cf. [Co2], [Cu2], [Pu], [EP]) if there is a k-tuple of commuting normal operators N = (N1 , · · · , Nk ) on a Hilbert space K, containing H as a subspace, satisfying Sj = Nj |H ,
j = 1, 2, · · · , k.
Without loss of generality, we may only consider the m.n.e.(minimal normal extension), i. e. there is no improper reducing subspace of N in K H. S is said to be pure, if there is no improper reducing subspace of S in H. Yakubovich [Y1], [Y2] called a subnormal operator with finite rank self-commutator as a subnormal operator of finite type, cf. also [X5], [X6], [X8]. Let us call a subnormal k-tuple of operators S = (S1 , · · · , Sk ) is of finite type if rank[Sj∗ , Sj ] < +∞, j = 1, 2, · · · , k. In this case, it is also that rank[Si∗ , Sj ] < +∞, i 6= j, i, j = 1, 2, · · · , k. In this paper we only study pure subnormal k-tuple of operators S of finite type on H.
2
Daoxing Xia
The main mathematical tool of this study is the analytic model of S in [X1], [X2], [X3], [X5]. Let M = ∨kj=1 [Sj∗ , Sj ]H. (1) Then dim M < +∞, since we only consider the operators of finite type. Let def Cij = [Si∗ , Sj ]|M as an operator on the finite dimensional space M, since M reduces
[Si∗ , Sj ].
def
Λj = (Sj∗ |M )∗ ∈ L(M ),
(2) Let (3)
since M is invariant with respect to Sj∗ , j = 1, 2, · · · , k (cf.[X1]). Let def
Rij (z) = Cij (z − Λj )−1 + Λ∗i ,
(4)
and def
Qij (w, z) = (w − Λ∗i )(z − Λj ) − Cij def
Pij (z, w) = det Qij (z, w).
(5) (6)
Then Pij (z, w) = Pji (w, z). The polynomial Pij (z, w) is with leading term z ν wν where ν = dim M . For A = (A1 , · · · , Ak ), let spjp (A∗ )∗ be the conjugate of the joint point spectrum, i. e. the set of all (w1 , · · · , wk ) ∈ Ck satisfying the condition that there is a vector f ∈ H, f 6= 0 such that A∗j f = wj f, j = 1, 2, · · · , k. In §3, some relation between the right spectrum of a pure subnormal tuple S of operators of finite type and spjp (S∗ )∗ is given. In §5, we introduce a union of domains S in some Riemann surfaces which is in a domain in an algebraic variety Sa . Those S and Sa are determined basically by {Pij (·, ·)} in (6). The aim of this paper is to determine a substantial part S of spjp (S∗ )∗ for pure subnormal k-tuple of operators S of finite type in Theorem 6.1. The principal current has been studied by Pincus and Carey [PC1], [PC2], [Pi]. In this paper, we will give the concrete form Theorem 7.1 of the principal current for the pure subnormal operators of finite type. Besides, in §7, we will give some operator ∗ U+ − I = 0 for the identities for S, which are the generalization of the identity U+ unilateral shift U+ .
2. Preliminaries In order to make the proofs in this paper readable, we have to list the some basic facts in the theory of the analytic model of subnormal operators of finite type which is a special case in [X1], [X2], [X3].
Subnormal tuple of operators
3
Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators of finite type on H with m.n.e. N. Let E(·) be the spectral measure of N on sp(N). Let PM be the projection from H to M, where M is defined in (1). Let def
e(·) = PM E(·)|M
(7)
ˆ be the Hilbert space complebe the L(M )-valued positive measure on sp(N). Let H Qk −1 tion of the span of all functions j=1 (λj − uj ) a for a ∈ M and λj ∈ ρ(Sj ), j = 1, 2, · · · , k, with respect to Z (f, h) = (e(du)f (u), h(u)). sp(N)
ˆ such that Then there is a unitary operator U from H onto H (U Sj U −1 f )(u) = uj f (u),
u = (u1 , · · · , uk ) ∈ sp(N)
and (U Sj∗ U −1 f )(u) = uj (f (u) − f (Λ)) + Λ∗j f (Λ)
(8)
where f (Λ) = sp(N) e(du)f (u). This (U S1 U −1 , · · · , U Sk U −1 ) is the analytic model of S. ˆ and U = I, i. e. we Without loss of generality, we may assume that H = H only have to consider the analytic model of S. For u ∈ σ(N), (9) Qij (u, u)e(du) = 0. R
Let def
P (S) = {u ∈ Ck : Pij (u, u) = 0, i, j = 1, 2, · · · , k} Then from (8) (cf. also [X3], [PX]), sp(N) ⊂ P (S). Let def
Z
µj (z) =
sp(N)
uj − Λj e(du), z ∈ ρ(Nj ). uj − z
Then µj (·) is analytic on ρ(Nj ) and µj (z) = 0 for z ∈ ρ(Sj ). Let T = (T1 , · · · , Tk ) be a k-tuple of commuting operators on H. If there is a finite dimensional subspace K ⊂ H such that [Ti∗ , Tj ]H ⊂ K, Ti∗ K ⊂ K for i, j = 1, 2, · · · , k. Then T is said to be of finite type. In this case let def
Cij = [Ti∗ , Tj ]|K ,
def
Λi = (Ti∗ |K )∗
for i, j = 1, 2, · · · , k. Define Rij (z) = Cij (z − Λj )−1 + Λ∗i and Qij (w, z) = (w − Λ∗i )(z − Λj ) − Cij as in §1. The following lemma is useful for §4 and the future study.
4
Daoxing Xia
Lemma 2.1
Let T be a k-tuple of operators of finite type. Then [Rmj (z), Rnj (z)] = 0, m, n, j = 1, 2, · · · , k,
(10)
for z ∈ ρ(Λj ). Proof Without loss of generality, we may assume that m = j = 1, n = 2. Define Ai = wi − Ti∗ , i = 1, 2, B1 = z − Ti for z ∈ ρ(T1 ), wi ∈ ρ(Ti ). Then [Ai , B1 ] = [Ti∗ , T1 ] = Ci1 PK , −1 −1 −1 −1 −1 where PK is the projection from H to K. Thus [A−1 i , B1 ] = Ai B1 Ci1 PK B1 Ai . Therefore −1 −1 −1 −1 −1 −1 −1 −1 PK A−1 1 A2 B1 |K = PK A1 [A2 , B1 ]|K + PK A1 B1 A2 |K .
(11)
The right-hand side of (11) equals −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 PK A−1 1 A2 B1 C21 PK B1 A2 |K + PK [A1 , B1 ]A2 |K + PK B1 A1 A2 |K −1 −1 −1 −1 −1 −1 −1 = PK A−1 (w2 − Λ∗2 )−1 + PK A−1 1 B1 C11 PK B1 A1 A2 |K 1 A2 B1 C21 (z1 − Λ1 )
+(z1 − Λ1 )−1 (w1 − Λ∗1 )−1 (w2 − Λ∗2 )−1 . since
A−1 i |K
= (wi − Λ∗i )−1 and
PK B1−1 |K
(12) = (z − Λ1 )−1 . On the other hand
−1 −1 −1 −1 −1 −1 −1 PK A−1 1 B1 |K = PK A1 B1 C11 PK B1 A1 |K + PK B1 A1 |K .
Thus −1 −1 PK A−1 (w1 − Λ∗1 )−1 ) = (z − Λ1 )−1 (w1 − Λ∗1 )−1 . 1 B1 |K (I − C11 (z − Λ1 )
Therefore −1 −1 . PK A−1 1 B1 |K = Q11 (w 1 , z)
(13)
From (11), (12) and (13), we have −1 −1 PK (A1 A−1 (w2 − Λ∗2 )−1 2 B1 )|K Q21 (w 2 , z)(z − Λ1 )
= (Q11 (w1 , z)−1 C11 + I)(z − Λ1 )−1 (w1 − Λ∗1 )−1 (w2 − Λ∗2 )−1 = Q11 (w1 , z)−1 (w2 − Λ∗2 )−1 Thus −1 −1 −1 = Q11 (w1 , z)−1 . PK (A−1 1 A2 B1 )|K Q21 (w 2 , z)(z − Λ1 )
Hence −1 −1 −1 PK (A−1 (w1 − R11 (z))−1 (w2 − R21 (z))−1 . 1 A2 B1 )|K = (z − Λ1 )
From [A1 , A2 ] = 0, we have [(w1 − R11 (z))−1 , (w2 − R21 (z))−1 ] = 0 which proves the lemma.
Subnormal tuple of operators
5
3. Right spectrum For a k-tuple of operators A = (A1 , · · · , Ak ) on Hilbert space H, let spr (A) = {(λ1 , · · · , λk ) ∈ Ck :
k X
(Aj − λj )(Aj − λj )∗ is not invertible }
j=1
be the right spectrum of A. It is obvious that spr (A) ⊃ spjp (A∗ )∗ . Proposition 3.1 Let S = (S1 , · · · , Sk ) be a subnormal k-tuple of operators of finite type on H, with m.n.e. N = (N1 , · · · , Nk ). If λ = (λ1 , · · · , λk ) satisfies λj ∈ σ(Sj ) ∩ ρ(Nj ), j = 1, 2, · · · , k and λ ∈ spr (S), then λ ∈ spjp (S∗ )∗ . Proof If (λ1 , · · · , λk ) ∈ spr (S), then there is a sequence {fm } ⊂ H satisfying kfm k = 1, such that k X
lim
m→∞
since
Pk
j=1 (Sj
(Sj − λj )(Sj − λj )∗ fm = 0,
j=1
− λj )(Sj − λj )∗ is self-adjoint. Thus
lim k(Sj − λj )∗ fm k = 0, j = 1, 2, · · · , k. P since k(Sj − λj )∗ fm k2 ≤ ( j (Sj − λj )(Sj − λj )∗ fm , fm ). On the other hand from (8) (cf. [X1], [X3]) m→∞
(Sj∗ − λj )fm (u) = (uj − λj )fm (u) − (uj − Λ∗j )fm (Λ), where
Z fm (Λ) =
e(du)fm (u). sp(N)
We may choose a subsequence {fmn } of {fm } such that fmn (Λ) → a ∈ M, as mn → ∞, since kfm (Λ)k ≤ 1. Therefore, from fmn (u) =
uj − Λ∗j u j − λj
fmn (Λ) +
1 (Sj∗ − λj )fmn uj − λj
and |uj − λj | ≥ dist(λj , σ(Nj )) > 0, for λj ∈ ρ(Nj ), it follows that as a sequence of vectors in H, {fmn } converges to g(u) =
uj − Λ∗j u j − λj
a, j = 1, 2, · · · , k.
On the other hand, we have fmn (Λ) = µj (λj )∗ fmn (Λ) +
Z
e(du) (Sj∗ − λj )fmn (u). uj − λj
(14)
6
Daoxing Xia
Thus a = µj (λj )∗ a, j = 1, 2, · · · , k. But kgk = lim kfmk k = 1, therefore g is an eigenvector of Sj∗ corresponding to λj , for j = 1, 2, · · · , k, i. e. λ ∈ spjp (S∗ )∗ .
4. Joint eigenvectors In order to study the joint eigenvectors for Sj∗ and Sk∗ , we have to establish the following lemma. Lemma 4.1 Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators on H with m.n.e. N = (N1 , · · · , Nk ). Let zl ∈ ρ(Nl ), l = 1, 2, · · · , n, n ≤ k and w ∈ ρ(Nj ), then Z n Y e(du)(uj − Λ∗j ) = (Rνl j (w)∗ − zl )−1 µj (w)∗ + n Y (uνl − zl )(uj − w) l=1 l=1 n Z X m=1
e(du) m Y (uνl − zl )
(I + (zm − Λνm )(Rνm j (w)∗ − zm )−1 )
n Y
(Rνp j (w)∗ − zp )−1 .
p=m+1
l=1
(15) Proof For the simplicity of notation, we assume that νl = l. Let us prove (15) by mathematical induction. First, let us prove it for n = 1. From (7) and Z e(du) = (I − µj (w)∗ )Qji (w, z)−1 − Qji (w, z)−1 µi (z). (ui − z)(uj − w) (cf. [X1], [X3]). In [X1], it proves this identity in the case of i = j, but the method in the proof also applies to the case i 6= j. Therefore, Z e(du)(w − Λ∗j ) = (I − µj (w)∗ )(z − R1j (w)∗ )−1 (u1 − z)(uj − w) − (z − Λ1 )−1 (w − Rj1 (z))−1 µ1 (z)(w − Λ∗j ), since (w − Λ∗j )(z − Rij (w)∗ ) = Qji (w, z), and (w − Rji (z))(z − Λi ) = Qji (w, z). But µj (z)Rij (z) = Rij (z)µj (z) (cf. [X1], [X3]). Again, in [X1] it only proves the case of i = j. But the proof may extend to the case of i 6= j as well. We have Z e(du)(w − Λ∗j ) (u1 − z)(uj − w) = (z − R1j (w)∗ )−1 (I − µj (w)∗ ) − (z − Λ1 )−1 µ1 (z)(z − Λ1 )(z − R1j (w)∗ )−1 . From −1
(z − Λ1 )
−1
µ1 (z) = (z − Λ1 )
Z +
e(du) u1 − z
Subnormal tuple of operators
7
we have (15) for n = 1. Suppose that (15) holds for n. Then from Lemma 2.1, Z Z Z e(du)(uj − Λ∗j ) e(du) e(du) = + (w − Λ∗j ) n+1 n+1 n+1 Y Y Y (ul − zl )(uj − w) (ul − zl ) (ul − zl )(uj − w) l=1
l=1
l=1
and Z
e(du)Qj(n+1) (w, zn+1 ) n+1 Y (ul − zl )(uj − w)
Z
e(du)(−(uj − Λ∗j )(un+1 − zn+1 ) − (uj − w)(zn+1 − Λn+1 ))
l=1
=
n+1 Y
(ul − zl )(uj − w)
l=1
Z =−
e(du)(uj − Λ∗j ) − n Y (ul − zl )(uj − w) l=1
it follows that Z
Z
e(du)(zn+1 − Λn+1 ) , n+1 Y (ul − zl ) l=1
e(du)(uj − Λ∗j ) n+1 Y
(ul − zl )(uj − w)
l=1
Z =
e(du) n+1 Y
(I − (zn+1 − Λn+1 )(zn+1 − R(n+1)j (w)∗ )−1 )
(ul − zl )
l=1
Z −
e(du)(uj − Λ∗j ) (zn+1 − R(n+1)j (w)∗ )−1 . n Y (ul − zl )(uj − w) l=1
which proves the lemma. Lemma 4.2 Under condition of Lemma 4.1, let wj ∈ σ(Sj ) ∩ ρ(Nj ), j = 1, 2. Suppose there are cν ∈ C, ν = 1, · · · , k and a vector a ∈ M satisfying µj (wj )∗ a = a, j = 1, 2
(16)
Rνj (wj )∗ a = cν a, j = 1, 2; ν = 1, 2, · · · , n.
(17)
and Let def
fj (u) =
uj − Λ∗j a, j = 1, 2 uj − w j
8
Daoxing Xia
Then fj (·) = f2 (·) as vectors in H and Sj∗ fj = wj fj , j = 1, 2. Proof Z
(18)
From (15), (16), (17) and Lemma 4.1, it is easy to calculate that Z e(du) e(du) f (u) = f2 (u), zl ∈ ρ(Nl ), l = 1, · · · , k. 1 k k Y Y (ul − zl ) (ul − zl ) l=1
l=1
e(du) ˆ of σ(N). Then by the Plemelj formula Let g(u) = on the continuous part L du1 X g(u)f2 (u) X g(u)f1 (u) = n n Y Y (ul − zl ) (ul − zl ) def
l=2
l=2
P for almost all points u1 in L1 = ∂σ(S1 ), where is ranging over a finite set of all points u in σ(N) with the same u1 . However zl is an arbitrary point in ρ(Nl ), l = 2, · · · , n. Therefore g(u)f1 (u) = g(u)f2 (u), for almost all u1 ∈ L1 . ˆ Similarly e({u})f1 (u) = e({u})f2 (u) Thus e(du)f1 (u) = e(du)f2 (u) for almost u ∈ L. for every point spectrum u in sp(N). Thus f1 (·) = f2 (·) as vectors in H. The formula (18) follows from (8) and (16).
5. Riemann surfaces and algebraic varieties associated with subnormal tuple of operators Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators of finite type with m.n.e. N = (N1 , · · · , Nk ). Let Oj ⊂ σ(Sj ) ∩ ρ(Nj ), j = 1, 2, · · · , k be non-empty open sets. Let Sˆj (·) be a univalent analytic function on Oj , j = 1, 2, · · · , k. Let O(O1 , Sˆ1 , · · · , Ok , Sˆk ) be defined as {z = (z1 , · · · , zk ) ∈ Ck : Pmn (Sˆm (zm ), zn ) = 0 for zj ∈ Oj , j = 1, · · · , k}. (19) Let S0 be the union of these neighborhoods {O(O1 , Sˆ1 , · · · , Ok , Sˆk )}. On each O(O1 , Sˆ1 , · · · , Ok , Sˆk ), every function f (zj ), considered as a function on z = (z1 , · · · , zj , · · · , zk ) is a local coordinate, where f is an analytic univalent function on Oj . Then S0 is a union of domains in some Riemann surfaces. Proposition 5.1 The continuum part of sp(N) ⊂ ∂S0 , the boundary of S0 . Proof Let γ be a small arc in σ(N). Let γj = {zj : (z1 , · · · , zj , · · · , zk ) ∈ γ} ⊂ σ(Nj ). Then there are domains Oj ⊂ σ(Sj )∩ρ(Nj ) and Dj ⊂ ρ(Sj ) such that Dj ∪ γj ∪Oj is a domain and γj ⊂ ∂Oj ∩∂Dj . From (9), Pij (z i , zj ) = 0, for (z1 , · · · , zk ) ∈ γ. There are analytic functions Sˆj (·) on Dj ∪ γj ∪ Oj satisfying Sj (z) = z, z ∈ γj . Actually these Sˆj (·) are branches of multivalued Schwarz functions associated with
Subnormal tuple of operators
9
those subnormal operators Sj (cf. [AS], [X5], [X6], [X7]), j = 1, 2, · · · , k. Thus Pji (Sˆj (zj ), zi ) = 0, (z1 , · · · , zk ) ∈ γ. Therefore γ ⊂ ∂O(O1 , Sˆ1 , · · · , Ok , Sˆk ). Pk Pk Suppose P (w) = j=0 pk−j wj and Q = j=0 qk−j wj are polynomials with constant coefficients pj and qi . Assume that p0 6= 0, q0 6= 0. Let def
R(P, Q) = det(rij )i,j=1,2,··· ,2k be the resolvent of P and Q, where rij = pj−i rij = qi−k−j
for 1 ≤ i ≤ j ≤ k, for 1 ≤ i − k ≤ j, k + 1 ≤ i ≤ 2k,
and rij = 0 for other pairs of i and j. It is well-known that if there is a common solution of P (·) = 0 and Q(·) = 0, then R(P, Q) = 0. Consider the algebraic variety def
Sa = ∩i6=j {(u1 , · · · , uk ) ∈ Ck : R(Pii (·, ui ), Pij (·, uj )) = 0}. It is easy to see the following. Lemma 5.1 S0 ⊂ Sa . Let γ be a small arc in the sp(N) satisfying the condition that for each j the mapping γ → γj = {uj ∈ C : (u1 , · · · , uj , · · · , uk ) ∈ γ} is one to one and γj is a simple arc (there is no node in γj ). For each γj there is a simply connected domain Oj ⊂ σ(Sj ) ∩ ρ(Nj ) such that γj is in the boundary of Oj and there is an analytic function Sˆj (·) on Oj such that Pjj (Sˆj (u), u) = 0, u ∈ Oj and the boundary value of Sˆj (·) satisfying Sˆj (z) = z, z ∈ γj , since Pjj (z, z) = 0, z ∈ γj . Let Oγ = O(O1 , Sˆ1 , · · · , Ok , Sˆk )
(20)
defined in (19) by these {Oj , Sˆj (·)}. Let S be the union of those component in S0 which contains some Oγ as a subset. This Sˆj (·) in (20) is a branch of the Schwarz function related to the subnormal operator Sj (cf. [X5], [X6], [X7], [Y1] and [Y2]). Thus the boundary of S must be also in sp(N). Therefore ∂S = sp(N). Theorem 5.1 Let S be a pure subnormal k-tuple of operators of finite type, then S ⊂ Sa . Let f be an analytic function on a neighborhood of σ(S1 )×· · ·×σ(Sk ). Define Z uj − Λj µj (z; f ) = e(du)f (u), z ∈ ρ(Nj ) σ(N) uj − z as in [X2] and [X3]. Then µj (·; f ) is an analytic function on ρ(Nj ) and µj (z; f ) = 0,
z ∈ ρ(Sj ).
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Daoxing Xia
By Plemelj formula, the boundary value of µj (·; f ) at the continuum part Lj ⊂ σ(Nj ) is X e(du) (uj − Λj ) f (u), duj P where is ranging over a finite set of points of u ∈ sp(N) with the same j-th coordinate uj . Therefore, there are L(M )-valued analytic functions µ ˆj (u) on S0 such that on the continuum part L of sp(N) µ ˆj (u) = (uj − Λj )
e(du) . duj
It is easy to see that (uj − Λj )−1 µ ˆj (u) = (uk − Λk )−1 µ ˆk (u)
duj , for u ∈ S. duk
(21)
Remark Let π be a linear function a1 u1 +· · ·+ak uk and Sπ = a1 S1 +· · · , +ak Sk (cf. [PZ]). We may choose k linear functions π1 , π2 , · · · , πk which are linearly independent on Ck such that if we replace S1 , · · · , Sk by Sπ1 , · · · , Sπk respectively then µ ˆj (u) = µj (uj ) for u ∈ S0 .
6. S and spjp (S∗ )∗ Let Mu be the range of µ ˆj (u)∗ for u ∈ S, which is well-defined by (21) and Mu 6= {0} except a finite set F ⊂ S. Theorem 6.1 Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators of finite type. Then there is a finite set F such that S \ F ⊂ spjp (S∗ )∗ .
(22)
For w ∈ S \ F , and a ∈ Mw \ {0}, the vector f (u) =
uj − Λ∗j a, j = 1, 2, · · · , k uj − w j
(23)
satisfies f 6= 0 and Sj∗ f = wj f, j = 1, 2, · · · , k. Proof
For u in the continuum part of sp(N), X e(du) µj (u)∗ = (uj − Λ∗j ). duj
(24)
(25)
By (7) and (25), we may prove that µ ˆj (u)∗ M ⊂ µj (u)∗ M for a.e. u ∈ σ(N). Thus (I − µj (uj )∗ )ˆ µj (u)∗ = 0 for a. e. u ∈ sp(N). Therefore (I − µj (uj )∗ )ˆ µj (u)∗ = 0 for u ∈ S, since µj (·) and µ ˆj (·) are analytic functions. Hence for a ∈ Mu , w ∈ S µj (wj )∗ a = a, for j = 1, · · · , k.
Subnormal tuple of operators
11
On the other hand, from (9), we have (Sν (wν ) − Rνj (wj )∗ )ˆ µm (w)∗ = 0, w ∈ γ
(26)
for w ∈ γ and j, ν, m = 1, 2, · · · , k where {Sj (·)} are those functions in the proof of Proposition 5.1, since Sν (wν ) = wν for w ∈ γ. Thus (26) holds good for w ∈ Oγ . By analytic continuation, (26) holds good for w ∈ S. Therefore (16) and (17) are satisfied if a ∈ Mw and cν = Sν (wν ). From Lemma 4.2, it follows (23) and (24) which proves (22) and hence Theorem 6.1. Conjecture : spjp (S∗ )∗ \ S is a finite set.
7. Principal current. The short introduction of the principal current related to the present work can be seen in the introduction of [PX] or [PZ] and the papers [CP1], [CP2], [Pi]. Let us first list the following two lemmas. For a compact set σ ⊂ Cn , let A(σ) be the algebra generated by analytic functions f on a neighborhood of σ and its conjugate. Lemma 7.1 [PX] Let S be a pure subnormal k-tuple of operators of finite type with m.n.e. N. Then Z 1 mf dh trace[f (S), h(S)] = 2πi L for f, h ∈ A(sp(N)), where L is the union of a finite collection of closed curves and is also the union of a finite collection of algebraic arcs such that sp(N) is the union of L and a finite set. Furthermore, m(u) is an integer valued multiplicity function which is constant on the irreducible pieces (simple closed curves) of L. In this lemma, e(du) m(u) = 2πi · trace((uj − Λj ) ), for u ∈ L. duj Thus we have the following Lemma 7.2 The function m(·) defined in Lemma 7.1 is the boundary value of the def
function m(u) = dim Mu = rank µ ˆj (u), u ∈ S. Lemma 7.3 [PZ] Suppose that S is a pure subnormal k-tuple of finite tuple with m.n.e. N. The principal current of S l(f dh) = i · trace[f (S), h(S)] can be represented as Z 1 X l(f dh) = ml f dh 2π Cl l
where {Cl } is the collection of cycles spess (S) and the weights ml are spectral multiplicity of N at any regular point ζ of Cl .
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Daoxing Xia
In this lemma, ml is the value of function m(u) for u ∈ Cl . By the fact that sp(N) is the boundary of S, Cartan’s formula Z Z dω = ω, σ
∂σ
Lemma 7.1-7.3 and the fact that the non-negative integer valued function m(·) is piece-wise constant. We have the following Theorem 7.1 Let S = (S1 , · · · , Sk ) be a pure subnormal k-tuple of operators of finite type. Then for f, h ∈ A(σ(S1 ) × · · · × σ(Sk )), Z 1 mdf ∧ dh, i · trace[f (S), h(S)] = 2π S where m(·) = rank µ ˆj (·). Thus m(·) can be considered as an extension of the concept of Pincus principal function from subnormal operators to subnormal k-tuple of operators.
8. Operator identities. It is well-known that the unilateral shift U+ satisfies the identity ∗ U+ U+ − I = 0.
(27)
This identity also characterizes the unilateral shift. In this section we will introduce some operator identities for the subnormal tuple of operators of finite type which are the generalization of (27). P Let p(w, z) = pmn wm z n be a polynomial and A and B be operators. We adopt the Weyl ordering X def p(A, B) = pmn Am B n . Theorem 8.1 Let S = (S1 , · · · , Sk ) be a subnormal k-tuple of operators of finite type. Let Pij (ui , uj ) be the polynomial defined in (6) and R(Pii (·, ui ), Pij (·, uj )) be the polynomial of ui , uj defined in §5. Then for i, j = 1, 2, · · · , k, Pij (Si∗ , Sj ) = 0,
(28)
R(Pii (·, Si ), Pij (·, Sj )) = 0.
(29)
and
Proof
Suppose Pij (w, z) =
X m,n
pmn wm z n .
Subnormal tuple of operators
13
Then by (9), Pij (ui , uj ) = 0 for (u1 , · · · , uk ) ∈ sp(N). Thus for f, g ∈ H X (Pij (Si∗ , Sj )f, g) = pmn (Si∗ m Sj n f, g) X = pmn (Sj n f, Si m g) Z X = pmn (e(du)uj n f (u), ui m g(u)) sp(N) Z = (e(du)Pij (ui , uj )f (u), g(u)) = 0, sp(N)
which proves (28). Similarly, we may prove (29), since R(Pii (·, ui ), Pij (·, uj )) = 0 for (u1 , · · · , uk ) ∈ sp(N).
References [AS] D. Aharonov and H. S. Sapiro, Domains on which analytic functions satisfy quadrature identities, J. Anal. Math. 30 (1976), 39–73. [Co1] J. B. Conway, Theory of Subnormal Operators, Math. Surv. Mon. 36 (1991). [Co2] J. B. Conway, Towards a functional calculus for subnormal tuples: the minimal normal extension, 138 (1977), 543–577. [CP1] R. W. Carey and J. D. Pincus, Principal currents, Integr. Equ. Oper. Theory 8 (1985), 614–640. [CP2] R. W. Carey and J. D. Pincus, Reciprocity for Fredholm operators, Integr. Equ. Oper. Theory 90 (1986), 469–501. [Cu1] R. E. Curto, Connections between Harte and Taylor spectra, Rev. Roum. Math. Pure Appl. 31 (1986), 1–31. [Cu2] R. E. Curto, Joint hyponormality: a bridge between hyponormality and subnormality, Pro. Symp. Pure Math. 5 (1990), 69–91. [EP] J. Eschmeier and M. Putinar, Some remarks on spherical isometries, in vol. ”Systems, Approximation, Singular Integral Operators and Related Topics”(A. A. Borichev and N. K. Nikolskii, eds), Birkhauser , Bassel et al., 2001, 271-292. [Pi] J. D. Pincus, The principal index, Proc. Symp. Pure Math. 51 (1990), 373–393. [Pu] M. Putinar, Spectral inclusion for subnormal n-tuples, proc. Amer. Math. Soc. 90 (1984), 405–406.
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[PX] J. D. Pincus, D. Xia, A trace formula for subnormal tuples, Integr. Equ. Oper. Theory. 14 (1991), 469–501. [PZ] J. D. Pincus, D. Zheng, A remark on the spectral multiplicity of normal extensions of commuting subnormal operator tuples, Integr. Equ. Oper. Theory 16 (1993), 145–153. [T1] J. L. Taylor, A joint spectrum for several commuting operators, Jour. Functional Analysis 6 (1970), 172–191. [T2] J. L. Taylor, The analytic functional calculus for several commuting operators, Acta Mathematica 125 (1970), 1–38. [X1] D. Xia, On the analytic model of a subnormal operator, Integr. Equ. Oper. Theory 10 (1987), 255–289. [X2] D. Xia, Analytic theory of subnormal operators, Integr. Equ. Oper. Theory 10 (1987), 890-903. [X3] D. Xia, Analytic theory of a subnormal n-tuple of operators, Proc. Symp. Pure Math. 51 (1990), 617–640. [X4] D. Xia, Trace formulas for a class of subnormal tuple of operators, Integr. Equ. Oper. Theory 17 (1993), 417–439. [X5] D. Xia, On pure subnormal operators with finite rank self-commutators and related operator tuples, Integr. Equ. Oper. Theory 24 (1996), 107–125. [X6] D. Xia, On a class of operators with finite rank self-commutators, Integr. Equa. Oper. Theory 33 (1999), 489–506. [X7] D. Xia, Trace formulas for some operators related to quadrature domains in Riemann surfaces, Integr. Equ. Oper. Theory 47 (2003), 123–130. [X8] D. Xia, On a class of hyponormal operators of finite type, to appear in Integr. Equ. Oper. Theory. [Y1] D. V. Yakubovich, Subnormal operators of finite type I, Xia’s model and real algebraic curves, Revista Matem. Iber. 14 (1998), 95–115. [Y2] D. V. Yakubovich, Subnormal operators of finite type II, Structure theorems, Revista Matem. Iber. 14 (1998), 623–689.
Daoxing Xia Department of Mathematics Vanderbilt University Nashville, TN 37240 USA e-mail: [email protected]
