Isosymmetric Linear Transformations on Complex ... - Semantic Scholar
UNIVERSITY OF CALIFORNIA, SAN DIEGO
Isosymmetric Linear Transformations on Complex Hilbert Space
A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics
by
. Mark Stankus
Committee in charge: Professor Jim Agler, Chair Professor J. William Helton Professor Daniel E. Wulbert Professor Anthony Sebald Professor David D. Sworder
1993
The dissertation of Mark Stankus is approved, and it is acceptable in quality and form for publication on micro film:
-~~
Professor Jim Agler, Chairperson
University of California, San Diego
1993
III
TABLE OF CONTENTS
Signature page . . . Table of Contents . Acknowledgements Vita,Publications & Fields Of Study. Abstract . 1
The Hereditary Functional Calculus, The Spectrum and Resolvent Inequal ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
III
IV
V
VI
Vll
1
2
Rosenblum's Theorem, von Neumann algebras, and roots of hereditary
polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14
3
Families, Maximal Invariant Subspaces and Several Examples of Roots "
23
4
The Elementary Operator Theory of Isosymmetries
39
5
Classification of Several Classes of Isosymmetries . . . . . . . . . . . , ..
48
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
70
IV
Acknowledgements I especially want to thank Professor Agler for his help and encouragement. Thanks are also due to Professor J. William Helton, Professor Daniel E. Wulbert, Professor Anthony Sebald and Professor David D. Sworder. Financial assistance during my graduate studies was provided by the UCSD Mathematics Department and by grants from the National Science Founda tion and the Air Force Office of Scientific Research. I am grateful for all the support that my friends and family gave me throughout graduate school. It was great to be able to rely on Carolyn Cross for friendship and understanding. Finally, let me thank Peter Stephens and Mark Yasuda for proofreading my thesis, Jerome G. Braunstein for helping me and answering my questions about TeX frequently and to Paul Van Wamelen for reminding me that brevity in a final defense is a virtue.
v
VITA
1987 1987-1991 1992 1985-1989 1989 1991 1993
B.s., Mathematics, Rensselaer Polytechnic Institute Minor in Computer Science Teaching Assistant, Department of Mathematics, University of California, San Diego. Computer Programmer, University Of California, San Diego. Computer Programmer, IBM (Summers) M.S., Mathematics, University Of California, San Diego. C.Phil., Mathematics, University of California, San Diego. Ph.D., Mathematics, University of California, San Diego.
PUBLICATIONS I'm-isometric Linear Transformations on Hilbert Space," coauthored with
Jim Agler, submitted February 1993.
Coauthoring a paper on hereditary polynomials and matrix theory with
Jim Agler and Bill Helton.
FIELDS OF STUDY Major Field: Mathematics Studies in Operator Theory Professor Jim Agler
VI
Abstract of the Dissertation
Isosymmetric Linear Transformations on Complex Hilbert Space
by
Mark Stankus
Doctor of Philosophy in Mathematics
University of California, San Diego, 1993
Professor Jim Agler, Chair
We explore the elementary operator theory of the equation 00
2:
(0.1)
cm,nT*nT m
=0
m,n=l
for Cm,n E C, Cm,n nonzero for only finitely many m, nand T a bounded linear transf()rmation on a complex Hilbert space in Chapters 1 and 2. We explore the equation T*2T - T*T 2 + T - T*
(0.2)
=0
in greater depth in Chapters 4 and 5. Chapter 1 explores the algebraic and C* -algebraic aspects of the equation (0.1) and both the spectral picture of and growth conditions on the resolvent ofthe operator T satisfying (0.1). Chapter 2 explores the implications of Rosenblum's Theorem to the study of (0.1). These implications are sufficient in some cases to completely classify a solution to (0.1) given information about the spectrum of T. Chapter 2 also recalls a few definitions and results from the theory of von Neumann algebras which will be used in the rest of the paper. Chapter 3 guarantees the existence of maximal invariant subspaces M for an operator T such that T restricted to M is a member of a fixed family of Vll
operators. This provides an approach to completely solving the equation (0.1) for T for certain choices of
em ,n.
In Chapters 4 and 5, we study operators T satisfying (0.2). These oper ators are termed isosymmetries. The results of Chapters 1, 2 and 3 do not solve equation (0.2). Chapter 4 gives the elementary operator theory of isosymmetries. Chapter 5 classifies several collections of isosymmetries. Indeed, if T is an isosymmetry and T is hyponormal, T is a contraction, Im(T) ~ 0 or Im(T) S; 0, then T is subnormal and the minimal normal extension of T has the same properties.
If T*T
~
1, then T is the restriction to an invariant subspace of a direct integral
of rank one perturbations of the unilateral shift. If the spectrum of T equals its boundry, then T has the form of a direct integral of 1 x 1 and 2 x 2 matrices. These constraints arise naturally from the analysis of Chapter 4.
Vlll
Chapter 1 The Hereditary Functional Calculus, The Spectrum and Resolvent Inequalities In this chapter, we introduce the notation of the paper, describe the Hereditary Functional Calculus [Agl] for elements of a C*-algebra and give the elementary operator theory for roots of a hereditary polynomial. Within this paper, all Hilbert spaces will be complex and separable. If 11. and K, are Hilbert spaces, we let £(11., K,) denote the bounded linear transformations from 11. to K,. We denote £(11., 'H) by £(11.). If p E C[x, V], then, for m 0, p"(m,n) will be complex numbers such that p(x,y) =
~
0 and n
~
L p"(m,n)ynxm. Note
m,n
that p"(m,n) is zero for all but finitely many m and n. Unless otherwise indicated, whenever m and n are indices of summation, m and n will range over the positive integers. Throughout this paper, whenever a summation is taken over an infinite index set (e.g., LiElad, all but finitely many summands are zero ({i E I: ai =f:. O} is finite). This fact will be used tacitly many times in this paper. If m and n are positive integers and R is a ring, let Rm,n denote m by n matrices with entries in
R. If C is a unital C*-algebra, c E C and p E C [x, y], then p( c) E C is defined 1
2
by
p(c) =
(1.1 )
~pA(m,n)CoonCm.
m,n
If nand K are positive integers, M k E cn,n and Pk E C[x,y] for every 1 ::; k ::; K
and P =
K
l: M k ® Pk
E cn,n ® C[x, y], then p(c) E cn,n ® C is defined by
k=l
K
p(c) = ~ M k ® Pk(C) k=l
where Pk(C) is defined as in (1.1). This functional calculus is termed the Hereditary
Functional Calculus in [Agl] and has been used to solve a number of problems [Agl,Ag2, Ag3, Ag4, Ag5, Ag6, Ag7, Ag-S, C-P, Ml, M2, HI]. In the following proposition, we record some elementary algebraic facts about the Hereditary Functional Calculus which are given in [Agl, Ag6, Ag-S]. For
P E C[x,y], let p V be the unique member of C[x,y] which satisfies the relation pVp',Ti) =p(Ti,>') Explicitly,pV(x,y)
for all
= l: pA(n,m)ynxm. m,n
Proposition 1.2. Let C be a COO-algebra and
C[x,y] into C is
(a) p
1--+
a
>',p E C.
C
E C. The map p
1--+
p(c) from
vector space homomorphism and the following hold.
p(c) is an algebra homomorphism if and only if c is normal.
(b) For p E C[x,y], p(c)OO = pV(c). (c) Ifp E C[x], q E C[x,y], s E Cry], and t(x,y) (1.3)
= s(y)q(x,y)p(x), then
t(c) = s(cOO)q(c)p(c) , where p(c) and s(cOO) are defined as in the Riesz Functional Calculus.
(d) Ifp E C[x], then the definition ofp(c) from the Hereditary Functional Calculus
3
and the Riesz Functional Calculus agree. (e) If D is a unital C"'-algebra and
(1.4)
7r :
C --+ D is a unital *-representation, then
7r(p(c)) = p(7r(c)).
In the following lemma, several spatial properties of the Hereditary Func tional Calculus are given for the special case when the C"'-algebra is £(1i). The lemma is proved implicitly in [Ag1]. Proposition 1.5. Let p E C[x, y]. If I is an index set, T OI E £(1i0l ) for each a E I and sup{ IITOIII : a E I}
0 such that
Ip(A, A)llIhll ~ MII(T - A)hll for h E H and A E K. In particular,
II(T _ A)-III
(1.16)
~ ~-
for A E K and A f/. a(T). Proof.
For A E C, m
~
0 and n
~
0, let Cmn(A) E C such that
p(x + A, y + "X) =
L: Cmn(A)ynxm . m,n
8
Let
M mn = SUP{ICmn(A)1 : A E K} and
Pm = sup{II(T - A)mll : A E K}. Since Cmn(A) and II(T - A)mll are continuous functions of A and K is compact,
M mn
' - >'01
,h> O.
and (>' - >'0)/1>' - >'01 E K
Let h Let C
= inf{lf(z)1
: z E K}. Since K is compact and K
n f-l( {O}) =
= 2/h and € > 0 be such that
I:
(1.19)
ICmnl€m+n-M:::; hj2.
m+n>M
If >. =P >'0, I>. - >'01 < Ip( >., "X) 1
>
€
I:
and (>. - >'0)/1>' - >'01 E K, then cmn (>'
- >'o)m("X - "Xo)n, -
> I>. - >'olMlf > I>. - >'oiM
>
I:
cmn (>. - >'o)m("X - "Xot
m+n>M
m+n==M
(I~ =~ol) 1- m+n>M I: Icmnll>' o
(h -
I:
>'olm+n
Icmnll>' - >.olm+n-M)
m+n>M
I>. - >'oiM /C .
This establishes (1.18) and completes the proof of Proposition 1.17. The following proposition is a variation of Theorem 6.3 in [R-R2] which, together with (1.16) and (1.18), proves that a root of a nonzero polynomial has hyperinvariant subspaces for each exposed arc of the spectrum. Since the proof of this variation differs from the proof of Theorem 6.3 only slightly, we will describe how to alter the proof of Theorem 6.3, rather than write out the complete proof. We begin with several definitions to be used in the hypothesis and proof of the proposition. A smooth Jordan arc is a one-to-one function z(t)
1) into C such that Jlz/dt 2 exists everywhere in (0,1). If K
= x(t) + iy(t) from (0, ~
C and K is compact,
then J is an exposed arc of K if there exists an open disk 'D such that 'D and J is a smooth Jordan arc.
nK
= J
10
Definition 1.20. Let T E £(1-£), C be an exposed arc of u(T), V be an open disk such that V
n u(T)
V such that V\C
=
= C and V l and V 2 be the connected open subsets of
V l U V 2 • Let
f
> 0, k be
C is exposed to order k from both sides at
Zo
a positive integer and
Zo
E C.
for T if there exists A > 0 and two
line segmen ts L l and L 2 con taining Zo such that L l \ {zo}
~ Vt,
L 2\ {Zo}
~
V 2 and
II(z-T)-lll~~ z - Zo
(1.21 ) for z E (L l U L 2)\{zo}.
Let us agree to say that, for k a positive integer, C is exposed to order k from both sides for T if for every Zo E C, C is exposed to order k from both sides at
Zo
for T.
Proposition 1.22. Let T E £(1-£), k > 0 and C be an exposed arc of u(T). Suppose that C is exposed to order k from both sides for T. T has a hyperinvariant subspace N such that u(TIN) ~ C-. As previously mentioned, the proof of the existence of a hyperinvariant
Proof.
subspace for T differs from the proof of Theorem 6.3 in [R-R2] only slightly. We list these differences below.
1. In the first paragraph, assume that tan
19'(x)1
A,B,pV(X) = (4)B.,A.,p(X*))*
wherever 1i and JC are Hilbert spaces, p E C[x, y], q E C[x, y], c E C, B E £(1i), X E £(1i, JC) and A E £(JC). The following proposition uses (2.2) to give a condition on q E C[x, y] such that (pq)(T)
= 0 implies p(T) = 0, where p E C[x, y] and T
E £(1i).
Proposition 2.6. Let p E C[x, y], q E C[x, y] and T E £(1i). If (pq)(T)
= 0 and
q(A,p) =I 0 for every A,p E u(T), then p(T) = O. Proof.
Since p(x,y)q(x,y) = L q"(m,n)ynp(x,y)x m , Proposition 1.2 implies m,n
that (pq)(T) = 4>T.,T,q(p(T)). Since (pq)(T) = 0,0 E U(4)T.,T,q) or p(T) = O. By (2.2), that fact that u(T*) = {j7 : p E u(T)} and the hypothesis of the proposition,
o ~ u( 4>T.,T,q).
Therefore, p(T) = 0 and the proof of Proposition 2.6 is complete.
In contrast to Proposition 2.6, the following proposition shows that if T is a root of pq and p(T) is both positive and invertible, then uap(T) Definition 1.14) rather than just uap(T)
~
is positive and p(T) is invertible, then u ap(T)
Let S
= p(T)t and
R
= STS-l.
~
Since
for m :2: 0 and n :2: 0,
L: q"(m, n)S*R*n R m,n
S*((pq)(T))S
- o.
= O,p(T)
A q and T is similar to a root of q.
T*np(T)T m = S* R*n R m S
S*q(R)S
(see
A pq as guaranteed by Proposition 1.12.
Proposition 2.7. Let p E C[x,y],q E C[x,y] and T E £(1i). If(pq)(T)
Proof.
~ Aq
m
S
16
Since S is invertible, q(R) = O. By Proposition 1.12, similar to R, O"ap(T)
= O"ap(R)
~
0"
ap(R)
~
Aqo Since T is
A q • This completes the proof of Proposition 2.7.
The following proposition and (2.2) prove that one can combine properties of a polynomial in C[x, y] and facts involving the spectrum of a root to show that certain invariant subspaces are actually reducing. Proposition 2.8. Let p E C[x, y], T E £(1i), {M,N} be a pair of complimentary invariant subspaces for T and PI, P2 E £(1i) be the idempotents defined by
P1(m +n) = m (2.9)
P2 (m
+ n)
mE M,n EN. =
n
If T has the block operator form
T=[~ ~]
(2.10)
with respect to the Hilb~rt space decomposition 1i = M EEl M.1 and p(T) = 0, then
(2.11)
E=BY-YC,
(2.12)
co,B,p(Y*)
=0
Bo,c,p(Y)
=0
and
(2.13) where Y
= PMP1\M.1
Proof.
By (2.9), PI and P2 have the block operator forms
(2.14)
and PM E £(1i) is the orthogonal projection of1i onto M.
Pl=[~ ~]
P,= [~
with respect to the Hilbert space decomposition 1i
-n
=M
EEl M.1.
17
Computation using (2.10) and (2.14) yields
N = { TJV =
k :k [-Yk]
{[
~
and
(E-C~Y)l] :lEM'}.
Since N is invariant for T, E - BY Now, for m
E M.l }
= -YC.
Therefore, (2.11) holds.
0, computation using (2.10) and (2.11) yields
(2.15)
T
m _
-
[Bm BmY - Ycm ] o cm
.
Therefore, computation using (2.15), (1.1) and (2.1) yields
(2.16) Since p(T)
p(T) = [
p(B) p(B)Y - . > a}. If T is finitely cyclic, tben tbe essential spectrum of T is a compact of R U an.
an.
If >. E C and (X>. - 1)(X - >.) = 0, then>. E R U
Proof.
Therefore, by
Proposition 1.12, the approximate point spectrum is a subset of R U
aO'(T)
~
O'ap(T)
~
R U
an,
an.
Since
O'(T) must have the form K U Y for some K and
Y as stated in the hypothesis. If T is finitely cyclic, then
0'
~
e(T)
R U n- by
Proposition 1.12. This completes the proof of Proposition 4.2. The following proposition follows from Propositions 1.15 and 1.17 and trivial algebraic manipulations. Proposition 4.3. Let T E £(1i), T be an isosymmetry and K be a compact
subset of C. Tbe following bolds. (a) Tbere exists M >
a sucb tbat, II
(4.4)
wbenever>. E K, >. ct. O'(T) and>. ct. R U an,
(T - >.)-1
Ib " _.
_
M
..
i
_
..
In addition, if >'0 E C and K is a subset of tbe unbounded sector
s = {>'o + reiD : r
?
a and 00 :5 0 :5 0d
wbere 00 E Rand 01 E R, tben tbe following bold. (b) If >'0 E R, >'0 =I 1, >'0 =I -1 and S f>
a sucb tbat,
(4.5)
nR
=
{>'o}, tben tbere exist C >
wbenever>. E K, >. ct. O'(T) and
II (>. - T)-1 11:5
a ' -
a and
>'0 1< f, tben
~
(c) If >'0 E an, >'0 =I 1, >'0 =I -1 and tbe intersection of Sand tbe tangent line to tbe unit circle at >'0 is {>'o}, tben tbere exist C > 0 and
wbenever>. E K, >. ct. O'(T) and 0 . - >'0 1
0 sucb tbat,
tben (4.5) bolds.
41
(d) If >"0 = 1 or >"0 = -1 and both SnR = {>"o} and sn {z : Re(z) = >"o} = {>"o}, then there exist C > 0 and
o .. -
>"0
1< f,
f
>
0 such that, whenever>.. E I..
tI.
u(T),
then
II
(4.6)
(>.. _T)-l
II~ ~- '-'
Before continuing, note that (4.5) cannot be used instead of (4.6) in Proposition 4.3(d) since the 2 x 2 matrix
T=
[>"0
o
b]
>"0
E C 2 ,2
is an isosymmetry for >"0 E {-I, I} and bE C and, for b =J. 0, this matrix satisfies
(4.6) but not (4.5). Since (yx - l)(y - x) can be written as y(yx -1) - (yx - l)x or y(y - x)x - (y - x), the isometry equation can be viewed in two different ways as illustrated by the following lemma. In the following lemma and throughout the paper, 6. T will denote T*T - 1 and Im(T) will denote ii(T - T*) for any operator
T. Lemma 4.7. Let T E £(1i). The following are equivalent.
(a) T is an isosymmetry. (b) T* Im(T)T
= Im(T).
(c) 6. T T = T*6. T • We may now exploit Rosenblum's Theorem and Lemma 4.7 to classify some isosymmetries based on the location of their spectrum.
Corollary 4.8. Let T E £(1i) and T be an isosymmetry. If u(T) ~ D or u(T) n
D- = 4>, then T is self-adjoint. If u(T) is contained in the open upper or open lower half plane, then T is
uni~ary.
42
Proof.
If CT(T) ~.
n-
or CT(T)
n n- =
0 and h E 1i be such that P X h = -tho We wish to show that
Proof.
h = O. Since P is positive, (Xh,h) = (Xh,PXh) . :s; O.
(5.13)
Since Re(X) 2: 0 and (Xh, h) E R, (Xh, h) = Re(Xh, h) = (Re(X)h, h) 2: O.
Thus, by (5.13), (PXh,Xh)
o.
= O.
Since P is positive, PXh
= 0 and h = -tPXh =
This completes the proof of Lemma 5.12. We now prove the following lemma which will playa key role in the first
proof of the classification of isosymmetries which are contractions. Lemma 5.14 1fT E £(1i), T is a finitely cyclic isosymmetry and T is a contrac-
tion, then (T*)nTn converges in the strong operator topology to a positive contraction P such that T* PT
(1 - P)(l Proof.
= P,
ker(1 - P)
= ker(1 -
T*T), (1 - P)(1 - T) 2: 0 and
+ T) 2: o. Since T is a contraction, 1 - T*T 2: 0 and T*nTn - T*(n+l)T n+t
= T*n(1 -
T*T)T n 2: 0
for all n 2: 1. Thus, 1 2: T*T 2: T*2T 2 2: ...
(5.15)
and T*nTn converges in the strong operator topology to a positive contraction P ([Be],Proposition 1.1). To see that T* PT
= P, note that for
(T* PTh, k)
=
h, k E 1i,
(PTh, Tk)
58
lim (T*nT(n+I)h , Tk)
n-+(X)
lim (T*(n+I)Tn+lh k)
n~oo
'
(Ph, k) . To see that ker(1-P)
= ker(1-T*T), note that since T is an
isosymmetry,
Lemma 4.27 implies that T has the block operator form
[~ ~]
T =
(5.16)
with respect to the Hilbert space decomposition 1-l = M EBM.L where M
= ker(1-
T*T), V E £(1-l), V is an isometry and V* E = O. Using the decomposition (5.16) yields
Thus, ker(l - T*T)
=M
~
p=[~ ~]. ker(1 - P). On the other hand, if h E ker(1 - P), then
(5.15) implies that
IIhll2 = (Ph, h) = nlim (T*nTnh, h) .....oo Ilhll = IIT*Thli and, since T P) = ker(1 - T*T).
Therefore, ker(1 -
::; (T*Th, h) ::;
Ilh11 2.
is a contraction, h E ker(1 - T*T). Thus,
Finally, let s be either 1 or -1. In order to show that (1- P)(1- sT) we will show that for all n
~
1,
(1 - T*nT n)(1 - sT)
(5.17)
~
0.
First, note that
(1 - yn x n)(1 - sx) -
(1-yx)
1- ynxn (1-sx) 1-yx n-l
-
(1 - yx)(L: yi xi)(1 - sx) i=O n-l
-
(1 - yx)(L: X 2i )(1 - sx) i=O
~
0,
59
modulo the ideal generated by (yx-1 )(y-x). Therefore, since T is an isosymmetry, Proposition 1.10 implies n-l
(1 - T*nT n )(l - sT) = (1 - T*T)(E T 2i)(1 - sT). i=O
Thus, by Lemma 4.7, (1 - T*nTn)(l - sT) is self-adjoint. Since T is an
isosymmetry, a contraction and finitely cyclic, Lemma
5.11 implies that T is essentially normal and and oAT) ~ [-1,1] U
(l-N*n Nn )(l-sN)
~
aD.
Since
0 for all normal operators N which are both contractions and
isosymmetries, the same is true, by Proposition 3.29, for subnormal isosymmetries which are contractions. By [B-D-F], there exist R E £(11.) and K E £(11.) such that R is a contractive subnormal isosymmetry, K is compact and T = R
+ J{.
Therefore, (1 - T*nTn)(l - sT) is a compact perturbation of the positive operator
(l-R*nRn)(l-sR). By Weyl's Theorem ([R-R2]' Theorem 0.10), if (l_!*nTn)(l_ sT) does not possess any negative eigenvalues, then (5.17) holds. Since T is a contraction, 1 - T*nTn
~
0 and Re(l - sT)
~
o.
Therefore, by Lemma 5.14,
(1- T*nTn)(l- sT) does not possess any negative eigenvalues. This completes the proof of Lemma 5.14. The following proposition classifies contractive isosymmetries. T E £(11.) will be called cyclic if there exists a vector I such that the smallest invariant subspace for T containing I is 11.. Proposition 5.18 If T E £(11.) , T is an isosymmetry, and T is a contraction, then T is unitarily equivalent to an operator of the form
(5.19)
(AEBU)
where A is self-adjoint, for AEB
u.
IIAII :::; 1 and U
1M
is unitary and M is a";' invariant subspace
60
Proof.
Since an operator T E £(1-£) is subnormal if and only if, for each nonzero
vector I E 1-£, T restricted to the smallest invariant subspace for T containing I is subnormal [C2], it suffices to assume that T is cyclic. We will first construct a self-adjoint operator A and a unitary operator U and then give an intertwining isometry. By Lemma 5.14, T*nTn converges in the strong operator topology to a positive contraction P. We first construct a Hilbert space !C and U E £(!C). Let !Co = ran(Pl/2)-. Since
IIPl/2ThIl2 = (T* PTh, h) = (Ph, h) = IIP1/2h112 , there exist an isometry V E £(!C o) satisfying the relation
V p 1 / 2 h = p 1 / 2 T h,
h E 1-£.
By [F], there exists a Hilbert space !C containing !Co and a unitary operator U E
£(!C) such that !Co is invariant for U and U I !Co = V. Now, let A o : ran(l - P)1/2
--+
ran(l - P)1/2 be defined by
A o(l - P)1/2 h = (1 _ P)1/2Th,
hE1-£.
A o is well-defined since ker(l - P)1/2 = ker(l - P) = ker(l - T*T) is invariant for
T by Lemma 4.27. The following two computations follow from Lemma 5.14 and prove that A o is symmetric and IIAol1 ~ 1.
2
11(1 - P)l/2hIl - (A o(1- P)1/2h, (1 - P)1/2h)
((1 - P)h, h) -((1 - P)1/2Th, (1 _ P)1/2h) -
((1 - P)(l - T)h, h)
> 0
(A o(l - P)l/2h, (1 - P)l/2h)
+ 11(1 -
((1 - P)I/2Th, (1 - P)l/2h)
P)l/2hI12
+((1 - P)h, h) ((1 - P)(T + l)h, h)
> 0 Since
IIAol1 :::;
1 and Ao is symmetric, AD extends by continuity to a self-
adjoint contraction A E £(J) where J Finally, if L : £(H)
---+
= (ran(l _
P)I/2)-.
£(J EEl K) is defined densely by
L(h)
=
pl/2h ] [ (1 _ P)I/2h '
then L is an isometry and U [
o
0]
A
Lh - LTh
=
[u0 A0] [ (1 -
-
U pl/2h - pl/2Th ] [ A(l - P)I/2 h - (1 - P)I/2Th
-
[~].
pl/2h ] [ pl/2Th ] P)l/2h (1 - P)l/2Th
This completes the proof of Proposition 5.18. We now give a second proof of Proposition 5.18. Proof 2 of Proposition 5.18 .
Let T be an isosymmetry which is a contraction.
Proposition 5.18 will follow from Proposition 3.29 once it has been shown that T is subnormal. By [Ag4], since T is a contraction, T is subnormal if and only if for each n
(5.20)
~
1,
(1 - yx t(T)
~
o.
62
Now,
(1 - yxt+2
(1- yxt(l- yx?
(1 - y x t(1- x 2? (1 - y2)(1 - yxt(1- x 2) modulo the ideal generated by (yx - 1 )(y - x). Thus, by Proposition 1.2,
(1 - yxt+2(T) = (1 - T*2)((1 - yxt(T))(l - T 2). Thus, we need only show (5.20) for n = 1 and n = 2. Since T is a contraction, (5.20) holds for n
= 1.
Since (1 - yx?(T) is self-adjoint, we need only show that
0"((1- T*T)(l- T2))
(5.21 )
Since (l-N* N)(1-N2)
~
n (-00,0) =
Isosymmetric Linear Transformations on Complex Hilbert Space
A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics
by
. Mark Stankus
Committee in charge: Professor Jim Agler, Chair Professor J. William Helton Professor Daniel E. Wulbert Professor Anthony Sebald Professor David D. Sworder
1993
The dissertation of Mark Stankus is approved, and it is acceptable in quality and form for publication on micro film:
-~~
Professor Jim Agler, Chairperson
University of California, San Diego
1993
III
TABLE OF CONTENTS
Signature page . . . Table of Contents . Acknowledgements Vita,Publications & Fields Of Study. Abstract . 1
The Hereditary Functional Calculus, The Spectrum and Resolvent Inequal ities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
III
IV
V
VI
Vll
1
2
Rosenblum's Theorem, von Neumann algebras, and roots of hereditary
polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14
3
Families, Maximal Invariant Subspaces and Several Examples of Roots "
23
4
The Elementary Operator Theory of Isosymmetries
39
5
Classification of Several Classes of Isosymmetries . . . . . . . . . . . , ..
48
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
70
IV
Acknowledgements I especially want to thank Professor Agler for his help and encouragement. Thanks are also due to Professor J. William Helton, Professor Daniel E. Wulbert, Professor Anthony Sebald and Professor David D. Sworder. Financial assistance during my graduate studies was provided by the UCSD Mathematics Department and by grants from the National Science Founda tion and the Air Force Office of Scientific Research. I am grateful for all the support that my friends and family gave me throughout graduate school. It was great to be able to rely on Carolyn Cross for friendship and understanding. Finally, let me thank Peter Stephens and Mark Yasuda for proofreading my thesis, Jerome G. Braunstein for helping me and answering my questions about TeX frequently and to Paul Van Wamelen for reminding me that brevity in a final defense is a virtue.
v
VITA
1987 1987-1991 1992 1985-1989 1989 1991 1993
B.s., Mathematics, Rensselaer Polytechnic Institute Minor in Computer Science Teaching Assistant, Department of Mathematics, University of California, San Diego. Computer Programmer, University Of California, San Diego. Computer Programmer, IBM (Summers) M.S., Mathematics, University Of California, San Diego. C.Phil., Mathematics, University of California, San Diego. Ph.D., Mathematics, University of California, San Diego.
PUBLICATIONS I'm-isometric Linear Transformations on Hilbert Space," coauthored with
Jim Agler, submitted February 1993.
Coauthoring a paper on hereditary polynomials and matrix theory with
Jim Agler and Bill Helton.
FIELDS OF STUDY Major Field: Mathematics Studies in Operator Theory Professor Jim Agler
VI
Abstract of the Dissertation
Isosymmetric Linear Transformations on Complex Hilbert Space
by
Mark Stankus
Doctor of Philosophy in Mathematics
University of California, San Diego, 1993
Professor Jim Agler, Chair
We explore the elementary operator theory of the equation 00
2:
(0.1)
cm,nT*nT m
=0
m,n=l
for Cm,n E C, Cm,n nonzero for only finitely many m, nand T a bounded linear transf()rmation on a complex Hilbert space in Chapters 1 and 2. We explore the equation T*2T - T*T 2 + T - T*
(0.2)
=0
in greater depth in Chapters 4 and 5. Chapter 1 explores the algebraic and C* -algebraic aspects of the equation (0.1) and both the spectral picture of and growth conditions on the resolvent ofthe operator T satisfying (0.1). Chapter 2 explores the implications of Rosenblum's Theorem to the study of (0.1). These implications are sufficient in some cases to completely classify a solution to (0.1) given information about the spectrum of T. Chapter 2 also recalls a few definitions and results from the theory of von Neumann algebras which will be used in the rest of the paper. Chapter 3 guarantees the existence of maximal invariant subspaces M for an operator T such that T restricted to M is a member of a fixed family of Vll
operators. This provides an approach to completely solving the equation (0.1) for T for certain choices of
em ,n.
In Chapters 4 and 5, we study operators T satisfying (0.2). These oper ators are termed isosymmetries. The results of Chapters 1, 2 and 3 do not solve equation (0.2). Chapter 4 gives the elementary operator theory of isosymmetries. Chapter 5 classifies several collections of isosymmetries. Indeed, if T is an isosymmetry and T is hyponormal, T is a contraction, Im(T) ~ 0 or Im(T) S; 0, then T is subnormal and the minimal normal extension of T has the same properties.
If T*T
~
1, then T is the restriction to an invariant subspace of a direct integral
of rank one perturbations of the unilateral shift. If the spectrum of T equals its boundry, then T has the form of a direct integral of 1 x 1 and 2 x 2 matrices. These constraints arise naturally from the analysis of Chapter 4.
Vlll
Chapter 1 The Hereditary Functional Calculus, The Spectrum and Resolvent Inequalities In this chapter, we introduce the notation of the paper, describe the Hereditary Functional Calculus [Agl] for elements of a C*-algebra and give the elementary operator theory for roots of a hereditary polynomial. Within this paper, all Hilbert spaces will be complex and separable. If 11. and K, are Hilbert spaces, we let £(11., K,) denote the bounded linear transformations from 11. to K,. We denote £(11., 'H) by £(11.). If p E C[x, V], then, for m 0, p"(m,n) will be complex numbers such that p(x,y) =
~
0 and n
~
L p"(m,n)ynxm. Note
m,n
that p"(m,n) is zero for all but finitely many m and n. Unless otherwise indicated, whenever m and n are indices of summation, m and n will range over the positive integers. Throughout this paper, whenever a summation is taken over an infinite index set (e.g., LiElad, all but finitely many summands are zero ({i E I: ai =f:. O} is finite). This fact will be used tacitly many times in this paper. If m and n are positive integers and R is a ring, let Rm,n denote m by n matrices with entries in
R. If C is a unital C*-algebra, c E C and p E C [x, y], then p( c) E C is defined 1
2
by
p(c) =
(1.1 )
~pA(m,n)CoonCm.
m,n
If nand K are positive integers, M k E cn,n and Pk E C[x,y] for every 1 ::; k ::; K
and P =
K
l: M k ® Pk
E cn,n ® C[x, y], then p(c) E cn,n ® C is defined by
k=l
K
p(c) = ~ M k ® Pk(C) k=l
where Pk(C) is defined as in (1.1). This functional calculus is termed the Hereditary
Functional Calculus in [Agl] and has been used to solve a number of problems [Agl,Ag2, Ag3, Ag4, Ag5, Ag6, Ag7, Ag-S, C-P, Ml, M2, HI]. In the following proposition, we record some elementary algebraic facts about the Hereditary Functional Calculus which are given in [Agl, Ag6, Ag-S]. For
P E C[x,y], let p V be the unique member of C[x,y] which satisfies the relation pVp',Ti) =p(Ti,>') Explicitly,pV(x,y)
for all
= l: pA(n,m)ynxm. m,n
Proposition 1.2. Let C be a COO-algebra and
C[x,y] into C is
(a) p
1--+
a
>',p E C.
C
E C. The map p
1--+
p(c) from
vector space homomorphism and the following hold.
p(c) is an algebra homomorphism if and only if c is normal.
(b) For p E C[x,y], p(c)OO = pV(c). (c) Ifp E C[x], q E C[x,y], s E Cry], and t(x,y) (1.3)
= s(y)q(x,y)p(x), then
t(c) = s(cOO)q(c)p(c) , where p(c) and s(cOO) are defined as in the Riesz Functional Calculus.
(d) Ifp E C[x], then the definition ofp(c) from the Hereditary Functional Calculus
3
and the Riesz Functional Calculus agree. (e) If D is a unital C"'-algebra and
(1.4)
7r :
C --+ D is a unital *-representation, then
7r(p(c)) = p(7r(c)).
In the following lemma, several spatial properties of the Hereditary Func tional Calculus are given for the special case when the C"'-algebra is £(1i). The lemma is proved implicitly in [Ag1]. Proposition 1.5. Let p E C[x, y]. If I is an index set, T OI E £(1i0l ) for each a E I and sup{ IITOIII : a E I}
0 such that
Ip(A, A)llIhll ~ MII(T - A)hll for h E H and A E K. In particular,
II(T _ A)-III
(1.16)
~ ~-
for A E K and A f/. a(T). Proof.
For A E C, m
~
0 and n
~
0, let Cmn(A) E C such that
p(x + A, y + "X) =
L: Cmn(A)ynxm . m,n
8
Let
M mn = SUP{ICmn(A)1 : A E K} and
Pm = sup{II(T - A)mll : A E K}. Since Cmn(A) and II(T - A)mll are continuous functions of A and K is compact,
M mn
' - >'01
,h> O.
and (>' - >'0)/1>' - >'01 E K
Let h Let C
= inf{lf(z)1
: z E K}. Since K is compact and K
n f-l( {O}) =
= 2/h and € > 0 be such that
I:
(1.19)
ICmnl€m+n-M:::; hj2.
m+n>M
If >. =P >'0, I>. - >'01 < Ip( >., "X) 1
>
€
I:
and (>. - >'0)/1>' - >'01 E K, then cmn (>'
- >'o)m("X - "Xo)n, -
> I>. - >'olMlf > I>. - >'oiM
>
I:
cmn (>. - >'o)m("X - "Xot
m+n>M
m+n==M
(I~ =~ol) 1- m+n>M I: Icmnll>' o
(h -
I:
>'olm+n
Icmnll>' - >.olm+n-M)
m+n>M
I>. - >'oiM /C .
This establishes (1.18) and completes the proof of Proposition 1.17. The following proposition is a variation of Theorem 6.3 in [R-R2] which, together with (1.16) and (1.18), proves that a root of a nonzero polynomial has hyperinvariant subspaces for each exposed arc of the spectrum. Since the proof of this variation differs from the proof of Theorem 6.3 only slightly, we will describe how to alter the proof of Theorem 6.3, rather than write out the complete proof. We begin with several definitions to be used in the hypothesis and proof of the proposition. A smooth Jordan arc is a one-to-one function z(t)
1) into C such that Jlz/dt 2 exists everywhere in (0,1). If K
= x(t) + iy(t) from (0, ~
C and K is compact,
then J is an exposed arc of K if there exists an open disk 'D such that 'D and J is a smooth Jordan arc.
nK
= J
10
Definition 1.20. Let T E £(1-£), C be an exposed arc of u(T), V be an open disk such that V
n u(T)
V such that V\C
=
= C and V l and V 2 be the connected open subsets of
V l U V 2 • Let
f
> 0, k be
C is exposed to order k from both sides at
Zo
a positive integer and
Zo
E C.
for T if there exists A > 0 and two
line segmen ts L l and L 2 con taining Zo such that L l \ {zo}
~ Vt,
L 2\ {Zo}
~
V 2 and
II(z-T)-lll~~ z - Zo
(1.21 ) for z E (L l U L 2)\{zo}.
Let us agree to say that, for k a positive integer, C is exposed to order k from both sides for T if for every Zo E C, C is exposed to order k from both sides at
Zo
for T.
Proposition 1.22. Let T E £(1-£), k > 0 and C be an exposed arc of u(T). Suppose that C is exposed to order k from both sides for T. T has a hyperinvariant subspace N such that u(TIN) ~ C-. As previously mentioned, the proof of the existence of a hyperinvariant
Proof.
subspace for T differs from the proof of Theorem 6.3 in [R-R2] only slightly. We list these differences below.
1. In the first paragraph, assume that tan
19'(x)1
A,B,pV(X) = (4)B.,A.,p(X*))*
wherever 1i and JC are Hilbert spaces, p E C[x, y], q E C[x, y], c E C, B E £(1i), X E £(1i, JC) and A E £(JC). The following proposition uses (2.2) to give a condition on q E C[x, y] such that (pq)(T)
= 0 implies p(T) = 0, where p E C[x, y] and T
E £(1i).
Proposition 2.6. Let p E C[x, y], q E C[x, y] and T E £(1i). If (pq)(T)
= 0 and
q(A,p) =I 0 for every A,p E u(T), then p(T) = O. Proof.
Since p(x,y)q(x,y) = L q"(m,n)ynp(x,y)x m , Proposition 1.2 implies m,n
that (pq)(T) = 4>T.,T,q(p(T)). Since (pq)(T) = 0,0 E U(4)T.,T,q) or p(T) = O. By (2.2), that fact that u(T*) = {j7 : p E u(T)} and the hypothesis of the proposition,
o ~ u( 4>T.,T,q).
Therefore, p(T) = 0 and the proof of Proposition 2.6 is complete.
In contrast to Proposition 2.6, the following proposition shows that if T is a root of pq and p(T) is both positive and invertible, then uap(T) Definition 1.14) rather than just uap(T)
~
is positive and p(T) is invertible, then u ap(T)
Let S
= p(T)t and
R
= STS-l.
~
Since
for m :2: 0 and n :2: 0,
L: q"(m, n)S*R*n R m,n
S*((pq)(T))S
- o.
= O,p(T)
A q and T is similar to a root of q.
T*np(T)T m = S* R*n R m S
S*q(R)S
(see
A pq as guaranteed by Proposition 1.12.
Proposition 2.7. Let p E C[x,y],q E C[x,y] and T E £(1i). If(pq)(T)
Proof.
~ Aq
m
S
16
Since S is invertible, q(R) = O. By Proposition 1.12, similar to R, O"ap(T)
= O"ap(R)
~
0"
ap(R)
~
Aqo Since T is
A q • This completes the proof of Proposition 2.7.
The following proposition and (2.2) prove that one can combine properties of a polynomial in C[x, y] and facts involving the spectrum of a root to show that certain invariant subspaces are actually reducing. Proposition 2.8. Let p E C[x, y], T E £(1i), {M,N} be a pair of complimentary invariant subspaces for T and PI, P2 E £(1i) be the idempotents defined by
P1(m +n) = m (2.9)
P2 (m
+ n)
mE M,n EN. =
n
If T has the block operator form
T=[~ ~]
(2.10)
with respect to the Hilb~rt space decomposition 1i = M EEl M.1 and p(T) = 0, then
(2.11)
E=BY-YC,
(2.12)
co,B,p(Y*)
=0
Bo,c,p(Y)
=0
and
(2.13) where Y
= PMP1\M.1
Proof.
By (2.9), PI and P2 have the block operator forms
(2.14)
and PM E £(1i) is the orthogonal projection of1i onto M.
Pl=[~ ~]
P,= [~
with respect to the Hilbert space decomposition 1i
-n
=M
EEl M.1.
17
Computation using (2.10) and (2.14) yields
N = { TJV =
k :k [-Yk]
{[
~
and
(E-C~Y)l] :lEM'}.
Since N is invariant for T, E - BY Now, for m
E M.l }
= -YC.
Therefore, (2.11) holds.
0, computation using (2.10) and (2.11) yields
(2.15)
T
m _
-
[Bm BmY - Ycm ] o cm
.
Therefore, computation using (2.15), (1.1) and (2.1) yields
(2.16) Since p(T)
p(T) = [
p(B) p(B)Y - . > a}. If T is finitely cyclic, tben tbe essential spectrum of T is a compact of R U an.
an.
If >. E C and (X>. - 1)(X - >.) = 0, then>. E R U
Proof.
Therefore, by
Proposition 1.12, the approximate point spectrum is a subset of R U
aO'(T)
~
O'ap(T)
~
R U
an,
an.
Since
O'(T) must have the form K U Y for some K and
Y as stated in the hypothesis. If T is finitely cyclic, then
0'
~
e(T)
R U n- by
Proposition 1.12. This completes the proof of Proposition 4.2. The following proposition follows from Propositions 1.15 and 1.17 and trivial algebraic manipulations. Proposition 4.3. Let T E £(1i), T be an isosymmetry and K be a compact
subset of C. Tbe following bolds. (a) Tbere exists M >
a sucb tbat, II
(4.4)
wbenever>. E K, >. ct. O'(T) and>. ct. R U an,
(T - >.)-1
Ib " _.
_
M
..
i
_
..
In addition, if >'0 E C and K is a subset of tbe unbounded sector
s = {>'o + reiD : r
?
a and 00 :5 0 :5 0d
wbere 00 E Rand 01 E R, tben tbe following bold. (b) If >'0 E R, >'0 =I 1, >'0 =I -1 and S f>
a sucb tbat,
(4.5)
nR
=
{>'o}, tben tbere exist C >
wbenever>. E K, >. ct. O'(T) and
II (>. - T)-1 11:5
a ' -
a and
>'0 1< f, tben
~
(c) If >'0 E an, >'0 =I 1, >'0 =I -1 and tbe intersection of Sand tbe tangent line to tbe unit circle at >'0 is {>'o}, tben tbere exist C > 0 and
wbenever>. E K, >. ct. O'(T) and 0 . - >'0 1
0 sucb tbat,
tben (4.5) bolds.
41
(d) If >"0 = 1 or >"0 = -1 and both SnR = {>"o} and sn {z : Re(z) = >"o} = {>"o}, then there exist C > 0 and
o .. -
>"0
1< f,
f
>
0 such that, whenever>.. E I..
tI.
u(T),
then
II
(4.6)
(>.. _T)-l
II~ ~- '-'
Before continuing, note that (4.5) cannot be used instead of (4.6) in Proposition 4.3(d) since the 2 x 2 matrix
T=
[>"0
o
b]
>"0
E C 2 ,2
is an isosymmetry for >"0 E {-I, I} and bE C and, for b =J. 0, this matrix satisfies
(4.6) but not (4.5). Since (yx - l)(y - x) can be written as y(yx -1) - (yx - l)x or y(y - x)x - (y - x), the isometry equation can be viewed in two different ways as illustrated by the following lemma. In the following lemma and throughout the paper, 6. T will denote T*T - 1 and Im(T) will denote ii(T - T*) for any operator
T. Lemma 4.7. Let T E £(1i). The following are equivalent.
(a) T is an isosymmetry. (b) T* Im(T)T
= Im(T).
(c) 6. T T = T*6. T • We may now exploit Rosenblum's Theorem and Lemma 4.7 to classify some isosymmetries based on the location of their spectrum.
Corollary 4.8. Let T E £(1i) and T be an isosymmetry. If u(T) ~ D or u(T) n
D- = 4>, then T is self-adjoint. If u(T) is contained in the open upper or open lower half plane, then T is
uni~ary.
42
Proof.
If CT(T) ~.
n-
or CT(T)
n n- =
0 and h E 1i be such that P X h = -tho We wish to show that
Proof.
h = O. Since P is positive, (Xh,h) = (Xh,PXh) . :s; O.
(5.13)
Since Re(X) 2: 0 and (Xh, h) E R, (Xh, h) = Re(Xh, h) = (Re(X)h, h) 2: O.
Thus, by (5.13), (PXh,Xh)
o.
= O.
Since P is positive, PXh
= 0 and h = -tPXh =
This completes the proof of Lemma 5.12. We now prove the following lemma which will playa key role in the first
proof of the classification of isosymmetries which are contractions. Lemma 5.14 1fT E £(1i), T is a finitely cyclic isosymmetry and T is a contrac-
tion, then (T*)nTn converges in the strong operator topology to a positive contraction P such that T* PT
(1 - P)(l Proof.
= P,
ker(1 - P)
= ker(1 -
T*T), (1 - P)(1 - T) 2: 0 and
+ T) 2: o. Since T is a contraction, 1 - T*T 2: 0 and T*nTn - T*(n+l)T n+t
= T*n(1 -
T*T)T n 2: 0
for all n 2: 1. Thus, 1 2: T*T 2: T*2T 2 2: ...
(5.15)
and T*nTn converges in the strong operator topology to a positive contraction P ([Be],Proposition 1.1). To see that T* PT
= P, note that for
(T* PTh, k)
=
h, k E 1i,
(PTh, Tk)
58
lim (T*nT(n+I)h , Tk)
n-+(X)
lim (T*(n+I)Tn+lh k)
n~oo
'
(Ph, k) . To see that ker(1-P)
= ker(1-T*T), note that since T is an
isosymmetry,
Lemma 4.27 implies that T has the block operator form
[~ ~]
T =
(5.16)
with respect to the Hilbert space decomposition 1-l = M EBM.L where M
= ker(1-
T*T), V E £(1-l), V is an isometry and V* E = O. Using the decomposition (5.16) yields
Thus, ker(l - T*T)
=M
~
p=[~ ~]. ker(1 - P). On the other hand, if h E ker(1 - P), then
(5.15) implies that
IIhll2 = (Ph, h) = nlim (T*nTnh, h) .....oo Ilhll = IIT*Thli and, since T P) = ker(1 - T*T).
Therefore, ker(1 -
::; (T*Th, h) ::;
Ilh11 2.
is a contraction, h E ker(1 - T*T). Thus,
Finally, let s be either 1 or -1. In order to show that (1- P)(1- sT) we will show that for all n
~
1,
(1 - T*nT n)(1 - sT)
(5.17)
~
0.
First, note that
(1 - yn x n)(1 - sx) -
(1-yx)
1- ynxn (1-sx) 1-yx n-l
-
(1 - yx)(L: yi xi)(1 - sx) i=O n-l
-
(1 - yx)(L: X 2i )(1 - sx) i=O
~
0,
59
modulo the ideal generated by (yx-1 )(y-x). Therefore, since T is an isosymmetry, Proposition 1.10 implies n-l
(1 - T*nT n )(l - sT) = (1 - T*T)(E T 2i)(1 - sT). i=O
Thus, by Lemma 4.7, (1 - T*nTn)(l - sT) is self-adjoint. Since T is an
isosymmetry, a contraction and finitely cyclic, Lemma
5.11 implies that T is essentially normal and and oAT) ~ [-1,1] U
(l-N*n Nn )(l-sN)
~
aD.
Since
0 for all normal operators N which are both contractions and
isosymmetries, the same is true, by Proposition 3.29, for subnormal isosymmetries which are contractions. By [B-D-F], there exist R E £(11.) and K E £(11.) such that R is a contractive subnormal isosymmetry, K is compact and T = R
+ J{.
Therefore, (1 - T*nTn)(l - sT) is a compact perturbation of the positive operator
(l-R*nRn)(l-sR). By Weyl's Theorem ([R-R2]' Theorem 0.10), if (l_!*nTn)(l_ sT) does not possess any negative eigenvalues, then (5.17) holds. Since T is a contraction, 1 - T*nTn
~
0 and Re(l - sT)
~
o.
Therefore, by Lemma 5.14,
(1- T*nTn)(l- sT) does not possess any negative eigenvalues. This completes the proof of Lemma 5.14. The following proposition classifies contractive isosymmetries. T E £(11.) will be called cyclic if there exists a vector I such that the smallest invariant subspace for T containing I is 11.. Proposition 5.18 If T E £(11.) , T is an isosymmetry, and T is a contraction, then T is unitarily equivalent to an operator of the form
(5.19)
(AEBU)
where A is self-adjoint, for AEB
u.
IIAII :::; 1 and U
1M
is unitary and M is a";' invariant subspace
60
Proof.
Since an operator T E £(1-£) is subnormal if and only if, for each nonzero
vector I E 1-£, T restricted to the smallest invariant subspace for T containing I is subnormal [C2], it suffices to assume that T is cyclic. We will first construct a self-adjoint operator A and a unitary operator U and then give an intertwining isometry. By Lemma 5.14, T*nTn converges in the strong operator topology to a positive contraction P. We first construct a Hilbert space !C and U E £(!C). Let !Co = ran(Pl/2)-. Since
IIPl/2ThIl2 = (T* PTh, h) = (Ph, h) = IIP1/2h112 , there exist an isometry V E £(!C o) satisfying the relation
V p 1 / 2 h = p 1 / 2 T h,
h E 1-£.
By [F], there exists a Hilbert space !C containing !Co and a unitary operator U E
£(!C) such that !Co is invariant for U and U I !Co = V. Now, let A o : ran(l - P)1/2
--+
ran(l - P)1/2 be defined by
A o(l - P)1/2 h = (1 _ P)1/2Th,
hE1-£.
A o is well-defined since ker(l - P)1/2 = ker(l - P) = ker(l - T*T) is invariant for
T by Lemma 4.27. The following two computations follow from Lemma 5.14 and prove that A o is symmetric and IIAol1 ~ 1.
2
11(1 - P)l/2hIl - (A o(1- P)1/2h, (1 - P)1/2h)
((1 - P)h, h) -((1 - P)1/2Th, (1 _ P)1/2h) -
((1 - P)(l - T)h, h)
> 0
(A o(l - P)l/2h, (1 - P)l/2h)
+ 11(1 -
((1 - P)I/2Th, (1 - P)l/2h)
P)l/2hI12
+((1 - P)h, h) ((1 - P)(T + l)h, h)
> 0 Since
IIAol1 :::;
1 and Ao is symmetric, AD extends by continuity to a self-
adjoint contraction A E £(J) where J Finally, if L : £(H)
---+
= (ran(l _
P)I/2)-.
£(J EEl K) is defined densely by
L(h)
=
pl/2h ] [ (1 _ P)I/2h '
then L is an isometry and U [
o
0]
A
Lh - LTh
=
[u0 A0] [ (1 -
-
U pl/2h - pl/2Th ] [ A(l - P)I/2 h - (1 - P)I/2Th
-
[~].
pl/2h ] [ pl/2Th ] P)l/2h (1 - P)l/2Th
This completes the proof of Proposition 5.18. We now give a second proof of Proposition 5.18. Proof 2 of Proposition 5.18 .
Let T be an isosymmetry which is a contraction.
Proposition 5.18 will follow from Proposition 3.29 once it has been shown that T is subnormal. By [Ag4], since T is a contraction, T is subnormal if and only if for each n
(5.20)
~
1,
(1 - yx t(T)
~
o.
62
Now,
(1 - yxt+2
(1- yxt(l- yx?
(1 - y x t(1- x 2? (1 - y2)(1 - yxt(1- x 2) modulo the ideal generated by (yx - 1 )(y - x). Thus, by Proposition 1.2,
(1 - yxt+2(T) = (1 - T*2)((1 - yxt(T))(l - T 2). Thus, we need only show (5.20) for n = 1 and n = 2. Since T is a contraction, (5.20) holds for n
= 1.
Since (1 - yx?(T) is self-adjoint, we need only show that
0"((1- T*T)(l- T2))
(5.21 )
Since (l-N* N)(1-N2)
~
n (-00,0) =
