ON CENTERED AND WEAKLY CENTERED OPERATORS
Vern Paulsen
Carl Pearcy
Srdjan Petrović
1. Introduction. Let H denote a separable, complex, Hilbert space and let L(H)
denote the algebra of all bounded linear operators on H. An operator T in L(H) is said
to be polynomially bounded (notation: T ∈ (PB)(H) or T ∈(PB)) if there exists an M ≥ 1
such that
kp(T )k ≤ M sup{|p(ζ)| : |ζ| = 1}
∀ polynomial p,
(1)
and to be power bounded (notation: T ∈ (PW)) if (1) holds for every polynomial of the
special form p(ζ) = ζ n where n is a positive integer. If T ∈ (PB) [resp., T ∈ (PW)], then
there is a smallest number M ≥ 1 which satisfies (1) [resp., (1) restricted]. This number
will be called the polynomial bound [resp., the power bound] of T and will be denoted by
Mpb (T ) [resp., Mpw (T )]. (If T ∈
/ (PB) [resp., T ∈
/ (PW)], we set Mpb [resp., Mpw ] equal
to +∞.) Also an operator T in L(H) is said to be similar to a contraction (notation:
T ∈ (SC)) if there exists an invertible operator S in L(H) such that kS −1 T Sk ≤ 1, and to
be completely polynomially bounded (notation: T ∈ (CPB)) if there exists an M ≥ 1 such
that one has
k(pij (T ))k ≤ M sup k(pij (ζ))k
|ζ|=1
∀n ∈ N, ∀ family {pij }ni,j=1 of polynomials,
1
(2)
where the operator (pij (T )) on the left side of (2) is an n × n matrix with operator entries
acting, in the usual fashion, on the direct sum of n copies of H, and (pij (ζ)) denotes the
obvious n × n complex matrix. If T ∈ (CPB), then there is, once again, a smallest number
M ≥ 1 satisfying (2), called the complete polynomial bound of T and denoted by Mcpb (T ).
It is elementary that
(SC) ⊂ (CPB) ⊂ (PB) ⊂ (PW),
and it was proved by the first author [13] that (SC)(H)=(CPB)(H) and that
Mcpb (T ) = min{kSkkS −1 k : kS −1 T Sk ≤ 1},
T ∈ (SC).
(3)
We will use this result throughout the paper without further comment.
On the other hand, it was shown by Foguel [6] (see also [7]) that (PB)6=(PW), and
it is a difficult and interesting open question, posed explicitly by Halmos in [8], whether
(PB) ⊂ (SC).
(4)
For more information concerning this circle of ideas, one might consult [14].
The purpose of this paper is to make a contribution toward determining the correctness of (4) along the following lines. In [17], the third author showed that if T ∈(PB)(H),
then T is the compression to a semi-invariant subspace of an operator T̃ acting on a Hilbert
space K ⊃ H such that T̃ ∈(PB)(K), the spectrum of T̃ is the unit circle in the complex
plane, and T̃ is weakly centered (notation: T ∈(WC)), meaning that T̃ T̃ ∗ commutes with
2
T̃ ∗ T̃ . (Said otherwise, T̃ is a dilation of T with certain nice properties.) It is an easy
consequence of this theorem (cf. [17] for details), that (4) is valid if and only if
(PB) ∩ (WC)in ⊂ (SC),
(5)
where (WC)in denotes the class of invertible weakly centered operators. This naturally
leads to the question: what is the structure of an invertible weakly centered operator? (It
should be said at once that weakly centered operators were studied briefly by Campbell in
[2] and [3] under the name binormal operators.) In what follows we study the structure
theory of the class (WC). Moreover, it turns out that there is another, related, class
of operators, called the centered operators, about which much is known. An operator
T ∈ L(H) is said to be centered (notation: T ∈(C)) if the doubly infinite sequence
{. . . , T n T ∗ n , . . . , T 2 T ∗ 2 , T T ∗ , T ∗ T, T ∗ 2 T 2 , . . . , T ∗ n T n , . . . }
consists of mutually commuting operators. It is obvious that (C) ⊂ (WC), and one may
thus ask the weaker question whether
(C) ∩ (PB) ⊂ (SC).
(6)
A partial answer to this question was given in [18], and one of the main purposes of this
paper is to complete the answer by establishing (6). To accomplish this, we first develop
some material about tensor products of operators in the classes (PW), (PB), and (CPB).
3
2. Centered operators. For use throughout the paper, we introduce the following
notation and terminology. We write C for the complex plane, D for the open unit disc in C,
and T for ∂D. As usual, N will denote the set of positive integers, N0 the set of nonnegative
integers, and Z the set of integers. If n ∈ N ∪ {ℵ0 } and K is any complex Hilbert space, we
write K(n) for the (orthogonal) direct sum of n copies of K. If T ∈ L(K) we write σ(T ) and
|σ(T )| for the spectrum and spectral radius of T , respectively. If {Dn }∞
n=1 is any bounded
sequence from L(K), we denote by Diag(D1 , D2 , . . . ) the operator in L(K(ℵ0 ) ) satisfying
Diag(D1 , D2 , . . . )(k1 , k2 , . . . ) = (D1 k1 , D2 k2 , . . . )
for all vectors (k1 , k2 , . . . ) in K(ℵ0 ) . Of course, Diag (D1 , D2 , . . . ) is also the direct sum
∞
⊕ Dn . Furthermore, if {Wn }∞
n=1 is any bounded sequence from L(K), we denote by
n=1
S{Wn } the operator in L(K(ℵ0 ) ) satisfying
S{Wn } (k1 , k2 , . . . , kn , . . . ) = (0, W1 k1 , W2 k2 , . . . , Wn kn , . . . ),
(7)
for all vectors (k1 , k2 , . . . ) in K(ℵ0 ) . (In other words, S{Wn } is the unilateral operatorweighted shift with weight sequence {Wn }.) In the special case in which all the weights in
(7) coincide with one weight W , we shall denote S{Wn } simply as S{W } . Clearly S{W } is
unitarily equivalent to the tensor product S ⊗ W acting on H ⊗ K in the usual way where
S is a unilateral shift in L(H) satisfying Sen = en+1 , n ∈ N, for some orthonormal basis
{en }∞
n=1 of H. We begin our program with the following lemma.
Lemma 2.1. Suppose T and N belong to L(H), where N is a normal operator of norm one.
Then T ∈(PB) [resp., T ∈(PW), T ∈(CPB)] if and only if the tensor product T ⊗N , acting
4
as usual on the Hilbert space H⊗H, belongs to (PB)(H⊗H) [resp., (PW), (CPB)] and one
also has Mpb (T ) = Mpb (T ⊗ N ) [resp., Mpw (T ) = Mpw (T ⊗ N ), Mcpb (T ) = Mcpb (T ⊗ N )].
Proof. Let A be the abelian unital C∗ - algebra generated by N , let X be the maximal
ideal space of A, and let ρ be the Gelfand transform of A onto C(X), which is, of course,
a C∗ -isomorphism. Let L(H) ⊗c A and L(H) ⊗c C(X) be the C∗ -tensor product algebras,
and recall (cf. [9, p. 848]) that there exists a C∗ -isomorphism ρ̃ of L(H) ⊗c A onto
L(H) ⊗c C(X) satisfying
ρ̃
n
X
k=1
Rk ⊗ Ak
!
=
n
X
Rk ⊗ ρ(Ak ),
k=1
for every n ∈ N and every pair of sequences {Rk }nk=1 ⊂ L(H) and {Ak }nk=1 ⊂ A. Moreover,
there exists another C∗ -isomorphism ϕ of L(H) ⊗c C(X) into the C∗ -algebra C(X, L(H))
of all continuous functions from X to L(H) under the supremum norm such that
ϕ
n
X
k=1
!
Rk ⊗ ρ(Ak )
= F,
where
F (x) =
n
X
(ρ(Ak )(x)) Rk
k=1
,
x ∈ X,
(cf. [9, p. 849]). We write G = (ϕ ◦ ρ̃)(T ⊗ N ), so G(x) = ω(x)T , x ∈ X, where ω = ρ(N ).
Clearly T ⊗ N ∈(PB) [resp., (PW), (CPB)] if and only if G is in the same class, and in this
case Mpb (T ⊗ N ) = Mpb (G) and similarly for the power bounds and complete polynomial
5
bounds. Suppose now that T ∈(PB), and let p be the polynomial p(ζ) =
kp(G)k = sup k
x∈X
n
X
k=0
k
ak G (x)k = sup k
x∈X
n
X
k=0
n
P
ak ζ k . Then
k=0
ak ω k (x)T k k.
(7)
For each x ∈ X, let qx be the polynomial
qx (ζ) =
n
X
ak ω k (x)ζ k ,
k=0
and note that from (7),
kp(G)k = sup kqx (T )k ≤ sup Mpb (T )kqx k∞ = Mpb (T ) sup sup |qx (ζ)| =
x∈X
x∈X
x∈X ζ∈T
= Mpb (T ) sup sup |
x∈X ζ∈T
n
X
k=0
ak ω k (x)ζ k |.
Since kN k = 1 by hypothesis, |ω(x)| ≤ 1 on X and thus
kp(G)k ≤ Mpb (T ) sup |
n
X
λ∈D− k=0
ak λk | = Mpb (T )kpk∞ .
Thus G is polynomially bounded and satisfies Mpb (G) ≤ Mpb (T ).
On the other hand, suppose now that G is polynomially bounded, let x0 ∈ X be
such that |ω(x0 )| = 1, and write ck = ak ω k (x0 ), k = 0, . . . , n. Then,
kp(T )k = k
n
X
k=0
k
ak T k = k
= sup k
x∈X
n
X
k=0
n
X
k=0
k
k
ck ω (x0 )T k ≤ sup k
x∈X
ck Gk (x)k = k
6
n
X
k=0
n
X
k=0
ck Gk k.
ck ω k (x)T k k =
Upon setting r(ζ) =
n
P
ck ζ k , we obtain
k=0
kp(T )k ≤ Mpb (G)krk∞ = Mpb (G) sup |
ζ∈T
= Mpb (G) sup |
ζ∈T
n
X
k=0
n
X
k=0
ck ζ k | =
k
ak (ω(x0 )ζ) | = Mpb (G)kpk∞ ,
which shows that T ∈ (PB) and that the polynomial bound of T satisfies Mpb (T ) ≤
Mpb (G). Thus all of the assertions of the lemma concerning polynomial boundedness are
true, and the arguments concerning power boundedness and complete polynomial boundedness follow from similar calculations, and are thus omitted.
Corollary 2.2. If T and U belong to L(H) with U unitary, then T ∈(PB) [resp.,
T ∈(PW), T ∈(CPB)] if and only if T ⊗ U ∈(PB) [resp., T ⊗ U ∈(PW), T ⊗ U ∈(CPB)],
and the corresponding bounds for T and T ⊗ U are the same.
We can now prove our first theorem.
Theorem 2.3. Suppose T and C belong to L(H), where kCk = |σ(C)| = 1. Then
T ∈(PB) [resp., T ∈(PW), T ∈(CPB)] if and only if T ⊗ C ∈(PB) [resp., T ⊗ C ∈(PW),
T ⊗ C ∈(CPB)], and furthermore the corresponding bounds for T and T ⊗ C are the same.
Proof. Suppose first that T ∈(PB). Since C is a contraction, there exists a Hilbert space
K ⊃ H and a unitary operator U ∈ L(K) such that H is semi-invariant for U and the
compression UH of U to H is C. It follows easily that H ⊗ H ⊂ H ⊗ K is a semi- invariant
subspace for T ⊗ U and that the compression (T ⊗ U )H⊗H = T ⊗ C. Since T ⊗ U ∈(PB)
by Corollary 2.2, and T and T ⊗ U have the same polynomial bound, it is clear that the
compression T ⊗ C ∈(PB) and satisfies Mpb (T ⊗ C) ≤ Mpb (T ).
7
To go the other way, suppose T ⊗ C ∈(PB). Since (PB) is invariant under multiplication by eiθ , θ ∈ R, there is no loss of generality in supposing that 1 ∈ σ(C). Since
T ∩ σ(C) ⊂ ∂σ(C), either 1 must be an eigenvalue of C or 1 ∈ σle (C), the left essential
spectrum of C. In the former case, T is the restriction to an invariant subspace of T ⊗ C,
and the conclusions that T ∈(PB) and Mpb (T ) ≤ Mpb (T ⊗ C) follow immediately. In the
latter case, there exists an orthonormal sequence {yn } in H such that kCyn − yn k → 0 and
hence
kC k yn − yn k → 0,
Now let p(ζ) =
m
P
k ∈ N.
(8)
ak ζ k be any fixed polynomial, and let be an arbitrary positive number.
k=1
Choose a unit vector x in H such that kp(T )xk > kp(T )k − /2. Now consider
Mpb (T ⊗ C)kpk∞ ≥ kp(T ⊗ C)(x ⊗ yn )k = k
=k
m
X
(ak T k x ⊗ C k yn )k ∼ k
k=1
m
X
k=1
m
X
k=1
!
ak (T k ⊗ C k ) (x ⊗ yn )k =
(ak T k x ⊗ yn )k = kp(T )x ⊗ yn k =
= kp(T )xk ≥ kp(T )k − /2,
so by virtue of (8) we have
Mpb (T ⊗ C)kpk∞ ≥ kp(T )k − provided n is chosen large enough. Since was arbitrary, this shows that T ∈(PB) and
that Mpb (T ) ≤ Mpb (T ⊗ C). Thus all of the conclusions of the theorem concerning polynomial boundedness have been established. The corresponding arguments concerning power
8
boundedness and complete polynomial boundedness are easy modifications of the above
arguments and are omitted.
The following corollary is very useful.
Corollary 2.4. If S and T belong to L(H) and S is a unilateral shift operator, then T
belongs to one of the classes (PB), (PW), (CPB), if and only if T ⊗ S [resp., S{T } (in the
notation introduced above)] belongs to the same class, and the corresponding bounds for
T and S{T } are identical.
As a first application of Corollary 2.4 we obtain the following interesting result.
Proposition 2.5. Let T ∈ L(H) and assume that there exists a sequence of operators
−1
−1
{An }∞
n=1 in L(H) satisfying kAn+1 T An k ≤ 1 and max{sup kAn k, sup kAn k} ≤ M . Then
n
n
there exists a single operator A in L(H) such that kA−1 T Ak ≤ 1 and kAk = kA−1 k ≤ M .
Proof. By [14, Theorem 8.1] and Corollary 2.4, it suffices to show that S{T } is similar to
a contraction and that Mcpb (S{T } ) ≤ M 2 . A calculation using the hypotheses shows that
−1
Diag(A−1
1 , A2 , . . . )S{T } Diag(A1 , A2 , . . . ) = S{A−1
n+1
Moreover S{A−1
n+1
T An }
T An } .
is obviously a contraction and
−1
2
kDiag(A−1
1 , A2 , . . . )k kDiag(A1 , A2 , . . . )k ≤ M ,
so Mcpb (S{T } ) ≤ M 2 .
The following is one of our main theorems and establishes a better result than (6).
9
Theorem 2.6. The inclusion (C) ∩ (PW) ⊂ (SC) is valid.
Proof. Let T be a centered and power bounded operator in L(H). Then, according to
the beautiful strucuture theorem of Morrel–Muhly [12], T can be written as a direct sum
T = TI ⊕ TII ⊕ TIII ⊕ TIV
where TN is a centered operator of type N , N = I, II, III, IV , according to the terminology of [12], and it is obvious that each TN ∈(PW) along with T . Moreover, the third
author showed [18, Theorem 1.1] that TI ⊕ TII ⊕ TIII ∈(SC), and thus it suffices to show
that TIV ∈(SC). The distinguishing feature of centered operators of type IV is that the
partial isometry appearing in the polar decomposition of such an operator is a unitary
operator. Thus we may write TIV = U P with U unitary, and it is a property of centered
operators [12, Lemma 3.1] that the sequence {U ∗ n P U n }∞
n=1 consists of mutually commuting (positive) operators. By Corollary 2.4, to show that TIV = U P ∈(SC), it suffices to
establish that S{U P } ∈(SC), and we know (Cor. 2.4) that S{U P } ∈(PW) with TIV . A
calculation shows that
Diag(1, U ∗ , U ∗ 2 , . . . )S{U P } Diag(1, U, U 2 , . . . ) = S{U ∗ n P U n }n∈N0 ,
and since S{U P } and S{U ∗ n P U n } are unitarily equivalent, it suffices to show that this
latter operator, which is obviously power bounded, belongs to (SC). But S{U ∗ n P U n } is
an operator weighted unilateral shift with mutually commuting normal weights, and thus
belongs to (SC) by [18, Theorem 1.2] or the stronger [16, Theorem 2.6].
10
Remark 2.7. The above theorem shows that for centered operators, power boundedness
is sufficient to imply similarity to a contraction, but this cannot be true for the larger class
of weakly cenetered operators. For, if (WC) ∩ (PW) ⊂ (SC), then, since every operator in
(PW) has a dilation that is in (WC) ∩ (PW) (cf. the construction in [17] or [15]), it would
follow that (PW) ⊂ (SC), which we know to be false.
Remark 2.8. We also remark that the construction in the proof of Theorem 2.6 shows that
it cannot be true that every power bounded operator S{Nn } , where {Nn } is a (bounded)
sequence of normal operators, belongs to (SC). (For if so, then S{U P } (and U P ) would
belong to (SC) for every invertible U P in (PW), which is false.) Similarly, it cannot be true
that every power bounded operator S{Wn } , where the (bounded) sequence {Wn } consists
of mutually commuting operators, belongs to (SC). (For if so, then every S{T } (and T )
that belongs to (PW) would also belong to (SC).) Thus, [16, Theorem 2.6] may be near
to best possible among theorems in which power boundedness implies similarity to
Theorem 2.9. If T is a centered, power bounded operator in L(H) and σ(T ) ⊃ T, then
either T is reflexive or T has a nontrivial hyperinvariant subspace. In particular, T has
nontrivial invariant subspaces.
Proof. By Theorem 2.6, T is similar to a contraction operator C (satisfying σ(C) ⊃ T).
By [4, Corollary 7.3], C is either reflexive or has a nontrivial hyperinvariant subspace, and
one knows that these properties are invariant under similarity transforms.
3. Weakly centered operators. In this section we study weakly centered operators. The following lemma is an easy consequence of the spectral theorem, and is contained
11
in [2].
Lemma 3.1. If T ∈ L(H) with polar decomposition T = U P (where P = (T ∗ T )1/2 ),
then T ∈(WC) if and only if P commutes with U P U ∗ . Furthermore, the class (WC)
is selfadjoint and closed under multiplication by complex numbers, taking inverses, and
formation of direct sums.
As an immediate consequence of Lemma 3.1, the earlier remark that to establish (4)
it suffices to establish (5), and the proof of Theorem 2.6, we have the following interesting
result.
Theorem 3.2. Every polynomially bounded operator in L(H) is similar to a contraction
if and only if every polynomially bounded operator-weighted shift S{Pn } whose weights
are positive semi-definite operators satisfying Pn Pn+1 = Pn+1 Pn , n ∈ N, is similar to a
contraction.
Proof. By (5) and Corollary 2.4, it is enough to consider S{T } with T weakly centered
and invertible. Let T = U P be the polar decomposition of T , let D be the unitary
operator Diag(I, U ∗ , U ∗ 2 , . . . ), and note that by Lemma 3.1, D−1 ST D = SPn where each
Pn is positive semi-definite and satisfies Pn Pn+1 = Pn+1 Pn .
To contrast the characterization of weakly centered operators in Lemma 3.1 with
that of centered operators, we recall from [12] the following.
Proposition 3.3. If T ∈ L(H) and is quasi- invertible with polar decomposition T = U P ,
then T is centered if and only if the infinite sequence {U n P U ∗ n }n∈N0 consists of mutually
commuting operators.
12
Under certain easily stated conditions, quasi-invertible weakly centered operators
are centered.
Proposition 3.4. If T ∈(WC) and is quasi- invertible (so that in the polar decomposition
T = U P , U is unitary), and U P U ∗ ∈ V, the unital von Neumann algebra generated by P ,
then T is centered (and of type IV).
Proof. Since V is abelian, it suffices to show that the family {U n P U ∗ n }n∈N0 is contained in V (because of Proposition 3.3). We argue by induction. By hypothesis, P and
U P U ∗ belong to V. Suppose now that for some n ≥ 1, {U k P U ∗ k }nk=0 ⊂ V, and let
B ∈ V. Then there is a net {pλ } of polynomials such that {pλ (P )} tends to B in the
weak operator topology (WOT). Thus U BU ∗ is the WOT-limit of the net {pλ (U P U ∗ )},
so U BU ∗ ∈ V. Thus, in particular, U n+1 P U ∗ n+1 = U (U n P U ∗ n )U ∗ ∈ V, and by induction
{U k P U ∗ k }k∈N0 ⊂ V.
Corollary 3.5. If T ∈(WC) and is quasi- invertible with polar decomposition T = U P ,
and P has a cyclic vector, then T is centered (and of type IV).
Proof. Since P has a cyclic vector, the unital von Neumann algebra V generated by P
is maximal abelian. Since P commutes with U P U ∗ by Lemma 3.1, U P U ∗ ∈ V, and the
result follows from Proposition 3.4.
This corollary was proved by S. Parrott (cf. [2, Theorem 6]).
Example 3.6. Unfortunately, Corollary 3.5 cannot be improved by replacing the hypothesis that P has uniform multiplicity 1 by the weaker hypothesis that P has uniform finite
multiplicity, as the following example shows.
13
Let K be a 4-dimensional complex Hilbert space, and let E = {ei }4i=1 be an ordered
orthonormal basis for K. Let P ∈ L(K) be the positive operator such that ME (P ), the
matrix for P relative to E, is

1


0



0

0
0
0
1
0
0
2
0
0
0



0

,

0

2
and let U1 and U2 be the unitary operators in L(K) such that

1


0

ME (U1 ) = 

0

0
0
0
0
1
1
0
0
0
0


√
1/ 2

 √


0
 1/ 2


 , ME (U2 ) = 


0
 0


0
1
√
1/ 2
0
√
−1/ 2
0
0
0
0



0

.

1 0

0
1
Define U = U1 U2 and T = U P . Then, as easy calculations show, P commutes with U P U ∗
so T ∈(WC), but P does not commute with U 2 P U ∗ 2 , so T ∈(C)
/
by Proposition 3.3.
The following curious property of weakly centered operators was proved in [3].
Proposition 3.7. If T ∈(WC)(H) and 0 is not in the interior of the numerical range of
T , then T is normal.
We would like to prove a theorem which completely determines the structure of
quasi-invertible weakly centered operators up to unitary equivalence, and in connection
with this project, we make the following conjecture.
Conjecture 3.8. Suppose U, P ∈ L(H), where U is a unitary operator, P is a positive semidefinite operator with trivial kernel, and P commutes with U P U ∗ . Then there
14
exist a maximal abelian von Neumann algebra A ⊂ L(H) such that P, U P U ∗ ∈ A and
a ∗ -automorphism Φ of A such that Φ(P ) = U P U ∗ .
We can establish this conjecture in some special cases.
Proposition 3.9. If the positive operator P generates a maximal abelian von Neumann
algebra or has pure point spectrum, then the conjecture is true for any pair (U, P ) where
U is unitary and U P U ∗ commutes with P .
Proof. If P generates a maximal abelian von Neumann algebra A, then U P U ∗ ∈ A since
U P U ∗ commutes with P , and it is elementary that one may define Φ(A) = U AU ∗ , A ∈ A,
to obtain an automorphism with the desired property.
Suppose now that P has pure point spectrum (i.e., the eigenvectors of P span H).
Then, since P is Hermitian, we can write P = ⊕ λi Ei , where the λi are all different and
i
the Ei are mutually orthogonal spectral projections for P with sum 1H . Since U P U ∗ is
unitarily equivalent to P , U P U ∗ also has pure point spectrum, and thus may be written
as ⊕ λj Fj , where the Fj (= U Ej U ∗ ) are the corresponding spectral projections of U P U ∗ .
j
Since P commutes with U P U ∗ (Lemma 3.1), the Ei and Fj all ommute, and consequently
there exists an ordered orthonormal basis E = {en }n∈N for H such that both ME (P ) and
ME (U P U ∗ ) are diagonal matrices. Since these matrices are unitarily equivalent, there
exists a permutation matrix Π such that ΠME (P )Π∗ = ME (U P U ∗ ). Let W be the unitary
operator in L(H) such that ME (W ) = Π, let A be the maximal abelian von Neumann
algebra consisting of all (normal) operators A ∈ L(H) such that ME (A) is diagonal, and
let Φ be the ∗ -automorphism of A defined by Φ(A) = W AW ∗ , a ∈ A. Then clearly
Φ(P ) = U P U ∗ , and the proof is complete.
15
For pairs (U, P ) for which Conjecture 3.8 is true, we obtain the desired structure
theorem.
Theorem 3.10. Suppose T = U P is a quasi-invertible operator in (WC) with U unitary
and P positive semidefinite, and suppose Conjecture 3.8 is true for the pair (U, P ). Let
A be the maximal abelian von Neumann algebra containing P and U P U ∗ given by Conjecture 3.8, and let Φ be a ∗ -automorphism of A such that Φ(P ) = U P U ∗ . Then there
exist
1)
a homeomorphism τ of X onto X, where X is the (compact, Hausdorff) maximal ideal space of A,
2)
a finite, regular, (perfect) Borel measure µ on X which satisfies µ◦τ ≡
µ, and
3)
a Hilbert space isomorphism W from H onto L2 (X, µ), such that
W T W −1 = SΘMΓ(P ) ,
(9)
where Γ is the Gelfand map of A onto C(X), MΓ(P ) is the operator
on L2 (X, µ) of multiplication by the function Γ(P ) in L∞ (X, µ), Θ is
a unitary operator on L2 (X, µ) that commutes with MΓ(P ) , and S is
the unitary operator on L2 (X, µ) defined by
(Sf )(x) = f (τ (x))(d(µ ◦ τ )/dµ)1/2 (x),
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f ∈ L2 (X, µ).
(10)
Furthermore, every operator of the form (9) where S, Θ, and Γ are as defined above is
weakly centered.
Proof. This proof is patterned after the proof of [12, Theorem 3]. Since the map
Γ : A → C(X) is a C∗ -algebra isomorphism of A onto C(X), it follows that ϕ = ΓΦΓ−1 is
a ∗ -automorphism of C(X), and hence there exists a homomorphism τ of X onto X such
that
[ϕ(f )](x) = f (τ (x)),
f ∈ C(X),
x ∈ X.
(11)
One knows (cf. [5, p.253, Proposition 3]) that there exists a unitary operator Ũ in L(H)
such that Φ(A) = Ũ A tildeU ∗ for all A in A, and hence from (11) we have
Γ(Ũ AŨ ∗ )(x) = Γ(A)(τ (x)),
A ∈ A,
x ∈ X.
(12)
Since A is maximal abelian, one knows (cf. [19, Lemma II.1.2]) that there exists a finite,
regular, perfect, Borel measure µ on X and a Hilbert space isomorphism W of H onto
L2 (X, µ) such that
W AW −1 = MΓ(A) ,
A ∈ A.
(13)
(To say that µ is perfect means that every equivalence class in L∞ (X, µ) contains an
element of C(X).) If we set Û = W Ũ W −1 and combine (12) and (13), we obtain
Û Mψ Û = Mψ◦τ ,
ψ ∈ C(X).
(14)
We now show that the measure µ ◦ τ on X is equivalent to the measure µ. To this end,
suppose first that K is a compact subset of X such that µ(K) = 0. Since µ is regular,
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there exists a sequence {Un }∞
n=1 of open subsets of X such that K ⊂ Un for each n and
µ(Un ) → 0. By an easy extension of Urysohn’s lemma (cf. [1, Problem 3V]), there exists
a decreasing sequence {gn } of nonnegative functions in C(X) such that χK ≤ gn ≤ χUn ,
n ∈ N. Thus, of course,
0≤
Z
2
2
gn |f | dµ ≤
X
Z
|f |2 dµ,
n ∈ N,
f ∈ L2 (X, µ).
Un
It follows easily from the absolute continuity of the integral that the sequence {Mgn } of
multiplication operators on L2 (X, µ) converges to zero in the strong operator topology,
and from (14) we deduce immediately that the sequence of operators {Mgn ◦τ k on L2 (X, µ)
also converges to zero in the strong operator topology. Since 0 ≤ χτ−1 (K) ≤ (gn ◦ τ )2 on
X, it follows that µ(τ −1 (K)) = 0 and thus that τ −1 maps compact sets of µ-measure zero
to compact sets of µ-measure zero. That τ −1 maps every Borel set E of µ-measure zero
to another such set now follows easily from the inner regularity of µ. Thus µ µ ◦ τ , and
a repetition of this argument with τ replacing τ −1 shows that µ ≡ µ ◦ τ . Consequently,
the oerator S given by (10) is well-defined, and a calculation shows that S is a unitary
operator on L2 (X, µ) satisfying
SMϕ S ∗ = Mϕ◦τ ,
ϕ ∈ L∞ (X, µ).
(15)
Upon setting Θ1 = S ∗ Û , we see from (14) and (15) that the unitary operator Θ1 commutes
with the maximal abelian algebra W AW ∗ on L2 (X, µ), and hence that Θ1 must have the
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form Θ1 = Mϕ1 , where |ϕ1 (x)| = 1 a. e. on X. Thus
W Ũ W −1 = Û = SΘ1 ,
so W T W −1 = W U P W −1 = (W U W −1 )(W P W −1 ) = W U W −1 MΓ(P ) =
(W Ũ W −1 )(W Ũ ∗ U W −1 )MΓ(P ) = SMϕ1 (W Ũ ∗ U W −1 )MΓ(P ) .
Since Ũ P Ũ = Φ(P ) =
U P U ∗ , the unitary operator Ũ ∗ U commutes with P , and hence the unitary operator
W Ũ ∗ U W −1 commutes with W P W −1 = MΓ(P ) . Thus the product Mϕ1 W Ũ ∗ U W −1 commutes with MΓ(P ) , and upon defining Θ = Mϕ1 W Ũ ∗ U W −1 , we see that (9) is true.
To prove the last statement of the theorem, it suffices by Lemma 3.1 to show that
SΘMΓ(P ) Θ∗ S ∗ commutes with MΓ(P ) . But
SΘMΓ(P ) Θ∗ S ∗ = SMΓ(P ) S ∗ = MΓ(P )◦τ ,
which obviously commutes with MΓ(P ) , so the theorem is proved.
Corollary 3.11. Suppose T ∈ L(H) is quasi-invertible with polar decomposition T =
U P , and P has pure point spectrum. Then T ∈(WC) if and only if there exist unitary
operators U1 , U2 in L(H) and there exists an ordered orthonormal basis E = {ei }∞
i=1 for
H such that
A)
T = U1 U2 P ,
B)
U2 commutes with P , and
C)
ME (P ) is diagonal and ME (U1 ) is a permutation matrix.
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Remark 3.12. Recall that the first operator T ∈(PW) \ (SC) was set forth by Foguel
[6] and Halmos [7], and Lebow [10] showed later that T ∈(PB).
/
We observe that the
same invariant used by Foguel–Halmos to show that T ∈
/ (SC) can be used to show that
T ∈
/ (PB), in view of the following proposition.
Proposition 3.13. Suppose T ∈ (PB)(H), and let Z(T ) = {x ∈ H : {T n x}∞
n=1 tends
weakly to zero}. Then
Z(T ) ∩ Z(T ∗ )⊥ = (0).
(16)
Proof. First note that if T satisfies (16) and S is invertible, then T̃ = ST S −1 also satisfies
(16) since Z(T̃ ∗ ) = S(Z(T )) and Z(T̃ ∗ )⊥ = S(Z(T ∗ )⊥ ), and recall from [11] that every
operator in (PB)(H) is similar to an operator of the form T̂ = Tac ⊕ Us acting on H =
Hac ⊕ Hs where Tac is absolutely continuous operator in (PB) and Us is a singular unitary
operator. Since Tac has a weak∗ continuous H ∞ functional calculus (cf., for example, [11])
and U is unitary, it is easy to see that Z(T̂ ) = Hac ⊕ Z(Us ) = Hac ⊕ Z(Us∗ ) = Z(T̂ ∗ ) so
certainly Z(T̂ ) ∩ Z(T̂ ∗ )⊥ = (0), and the proof is complete.
Remark 3.14. We close with a couple of remarks concerning [3]. Campbell’s form (I2 )
in Theorem 7 is superfluous — the matrix given there is unitarily equivalent to


0
y
x
0


with xy = 1 and x2 + y 2 = 2 + b2 , so is of the form (I3 ). He also makes a misstatement
on p.361 that 0 lies in the convex hull of the spectrum for all operators in the classes (I1 ),
(I2 ), (I3 ), etc.
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Department of Mathematics
Department of Mathematics
University of Houston
Texas A. & M. University
Houston, TX 77204
College Station, TX 77843
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