数理解析研究所講究録
第 1596 巻 2008 年 135-139

135

ON STAR MOMENT SEQUENCE OF
OPERATORSSun Hyun Park

Department of Mathematics, College of Natural Sciences,
Kyungpook National University,
Taegu 702-701, Korea
E-mail: sm1907s4@hanmail.net

AbstractLetbe a separable, infinite dimensional, complex Hilbert space. We call “anoperator $T$ acting on
has a star moment sequence supported on a set $K$ ’ when
there exist nonzero vectors and in
and a positive Borel measure such thatfor all $j,$ $k\geq 0$ . We obtain a characterization to find arepresenting star moment measure and discuss some related properties.
$\mathcal{H}$$\mathcal{H}$$u$$v$$\mathcal{H}$$\mu$$\langle T^{j}T^{k}u,v\rangle=\int_{K}\overline{z}^{j}z^{k}d\mu$

1. Introduction and Preliminaries

The results of this article will be appeared in other journal. And so we omit somebedetail proof here. Let be a (real or complex) Banach space, and denoted by
be a dual space of X. Following
the algebra of all bounded linear operators on X. Let[AG], we say that ain
has a moment sequence if there exist nonzero vectors
ofandand a positive Borel measure supported on the spectrum(and, of course,
if is a real Banach space) such that$\mathcal{L}(\mathcal{X})$$\mathcal{X}$$\mathcal{X}^{*}$$T$$x\in \mathcal{X}$$\mathcal{L}(\mathcal{X})$$\sigma(T)$$y\in \mathcal{X}^{*}$$\sigma(T)\subset \mathbb{R}$$T$$\mathcal{X}$$y(T^{n}x)= \int_{\sigma(T)}\lambda^{n}d\mu_{x,y}$,$n\in N_{0}$,(1.1)’2000 Mathematics Subject Classification. Primary $47A15,44A60$ , Secondary $47B20$ .
\dagger Key wods and phrases: moment sequence, invariant subspace, essentially normal operator, subnormaloperator.

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denotes, in usual, the set of nonnegative integers.
whereAtzmon and Godefroy then showed in [AG] that if is real and satisfies some addithat has a moment sequence (as in (1.1)))
tional conditions, that every operator inhas, in fact, a nontrivial invariant subspace. This immediately raises the following question.
Question 1.1. Let
be a separable, infinite dimensional, complex Hilbert space.
Which classes of operators have moment sequences?
$N_{0}$$\mathcal{X}$$\mathcal{L}(\mathcal{X})$$\mathcal{H}$Of course, at present, it is not known, in this context, whether having a moment
sequence implies the existence of invariant subspace. This led the authors of [FJKP] and, and they showed[CJKP] to undertake a study of this equation in the case that
do have moment sequences.
that various classes of operators inIn this note we consider a related question and define a new definition of momentsequence.
be a separable, infinite dimensional, complex Hilbert space.
Definition 1.2. Let
if there exist
has $a*$ -moment sequen ce supported onAn operator
thatsuchmeasurepositive
Borel
nonzero vectors and in and a$\mathcal{X}=\mathcal{H}$$\mathcal{L}(\mathcal{H})$$\mathcal{H}$$K\subset \mathbb{C}$$T\in \mathcal{L}(\mathcal{H})$

$u$

$v$

$\mathcal{H}$$\mu$$\langle T^{*j}T^{k}u, v\rangle=\int_{K}\overline{z}^{j}z^{k}d\mu$,$j_{)}k\in N_{0}$.

(1.2)

in almost cases.)In this note, we will establish some necessary and sufficient conditions for which an
has $a*$ -moment sequence, and in condition which equivalent tooperator
inmoment sequence.
(Throughout this note $K$ will be$T$$\sigma(T)$$*-$$\mathcal{L}(\mathcal{H})$2. Some Results. Then obviously $N$ has-moment sequence.
Let $N$ be a normal operator inMoreover, this fact can be improved to the case of subnormal operator as following.
have a -moment sequence
Proposition 2.1. Every subnormal operator in$a^{*}$$\mathcal{L}(\mathcal{H})$

$*$

$S$

on

$\sigma(S)$$\mathcal{L}(\mathcal{H})$.

We now characterize operators havingof this note.
. ThenTheorem 2.2. Let
if and only if there exist nonzero vectors$a^{*}$-moment sequence, whicf is the main theo-$rem$$T\in \mathcal{L}(\mathcal{H})$

$T$
$u$

has a -moment sequence supported on
andand insuch that$*$$v$$|\langle p(T^{*}, T)u, v\rangle|\leq\langle u, v\rangle\Vert p\Vert_{\infty}$

where$\Vert p\Vert_{\infty}$$(:= \sup_{z,5\in\sigma(T)}|p(z,\overline{z})|$for$p\in C[\overline{z}, z]$$| \langle p(T^{*},T)u, v\rangle|=\sum a_{jk}T^{*j}T^{k}$for,$\sigma(T)$$\langle u, v\rangle\geq 0$$\mathcal{H}$$p\in \mathbb{C}[\overline{z}, z]$,) is the sup norm supported on$p( \overline{z}, z)=\sum a_{jk}\overline{z}^{j}z^{k},$$j,$ $k\in N_{0}$$\sigma(T)$.

and137(Note that $p(T^{*},$ $T) \equiv\sum a_{jk}T^{*j}T^{k}$ which is not equal to $\sum a_{jk}T^{k}T^{*j}$ in general.)Sketch of Proof. Suppose
has $a*$ -moment sequence supported onThen there exist nonzero vectors and and a positive Borel measure such that$T\in \mathcal{L}(\mathcal{H})$

$u$

$v$

$\sigma(T)$.

$\mu$$\langle T^{sj}T^{k}u, v\rangle=\int_{\sigma(T)}\overline{z}^{j}z^{k}d\mu$,$j,$ $k\in N_{0}$.

For $j=k=0$ , we have$\langle u, v\rangle=\int_{\sigma(T)}1d\mu=\mu(\sigma(T))$.

Then$|\langle p(T^{*},\backslash T)u, v\rangle|$$=$$| \int_{\sigma(T)}p(\overline{z}, z)d\mu|\leq l_{(T)}^{|p(\overline{z},z)|d\mu}$$\leq$$\Vert p\Vert_{\infty}|\int_{\sigma(T)}1d\mu|=\Vert p\Vert_{\infty}\mu(\sigma(T))=||p\Vert_{\infty}\langle u, v\rangle$.

Conversely, without loss of generality we may assume that there exist nonzero vectors
$u,$

$v$

such that $\Vert u\Vert=||v||=1,$$\langle u, v\rangle\geq 0$$|(p(T^{*}, T)u,$Define$\tau$:$\mathbb{C}[\overline{z}, z]arrow \mathbb{C}$$v\rangle$

and$p\in C[\overline{z}, z]$$|\leq\langle u, v\rangle\Vert p\Vert_{\infty},$.

by$\tau(p(\overline{z}, z))=(p(T^{*}, T)u,$$v\rangle$.

Then is obviously linear and. So is positive. By the Hahn Banachtheorem, there exists a continuous linear mapping:
such that$=\tau(1)$, which implies that
andis positive. By Riesz representation
Iltheorem, there exists a positive Borel measure supported onsuch that$\tau(1)=\langle u, v\rangle\geq 0$$\tau$$\tau$$\tau_{\epsilon xt}$$||\tau_{\epsilon xt}||=||\tau$$\tau$$C(\sigma(T))arrow \mathbb{C}$$\tau_{ext}$$\sigma(T)$$\mu$$\tau_{ext}(p(\overline{z}, z))=\int_{\sigma(T)}p(\overline{z}, z)d\mu$Take$p(\overline{z}, z)=\overline{z}^{i}z^{j},$$i,j\in N_{0}$.

and since$\langle p(T^{*}, T)u, v\rangle=\tau(p(\overline{z}, z))=\tau_{ext}(p(\overline{z}, z))$we have that$\langle T^{*j}T^{k}u, v\rangle=\int_{\sigma(T)}\overline{z}^{j}z^{k}d\mu,$Hence$T$

has$a^{*}$-moment sequence supported on$\sigma(T)$$j,$ $k\in N_{0}$.

$\blacksquare$.

,$\tau_{\epsilon xt}|_{C[\overline{\iota},*]}=$

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Corollary 2.3. If an operator $T$ has a -moment sequence on.
has a -moment sequence supported on$*$$\tau*$$*$$\sigma(T)\subset \mathbb{C}$, then also$\overline{\sigma(T)}=\sigma(T^{*})$

The following corollary follows immediately from Theorem 2.2 and Corollary 2.3.
has a ’-moment sequence supported on
Corollary 2.4. If an operator
is a polynomial, then $p(T)$ has a -moment sequence supported on $p(\sigma(T))\cup$and$T$$\sigma(T)\subset \mathbb{C}$$*$$p(\overline{z}, z)$$\overline{p(\sigma(T))}$.

andRemark 2.5. If an operator has two nonzero invariant subspaces(Indeed,takeC.
, then has-moment sequence supported on. Then the $zero$ measure satisfiesandand $v\in N$ . Then
$T$$a^{*}$$T$$\mathcal{M}\perp \mathcal{N}$$\mathcal{M}$$\sigma(T)\subset$$T^{j}v\in \mathcal{N}$$T^{k}u\in \mathcal{M}$$\mathcal{N}$with$u\in \mathcal{M}$$\mu$$\langle T^{*j}T^{k}u, v\rangle=\langle T^{k}u, T^{j}v\rangle=0=\int_{\sigma(T)}\overline{z}^{j}z^{k}d\mu,$$j,$ $k\in N_{0}$. So we have this remark.)
The following Proposition 2.6 improves Proposition 1.6 in [FJKP].
are sequences inand. Suppose that
Proposition 2.6. Let
and , respectively, and for every $n\in N_{0}$ there
converging in norm to nonzero vectors
such that
suppo rted on the compact set
exists a Borel measure

for any set$\sigma(T)$$\{v_{n}\}$$\{u_{n}\}$$T\in \mathcal{L}(\mathcal{H})$$u_{0}$$\mathcal{H}$$v_{0}$$\sigma(T)\subset \mathbb{C}$$\mu_{\mathfrak{n}}$$\langle T^{rj}T^{k}u_{\mathfrak{n}}, v_{n}\rangle=\int_{\sigma(T)}\overline{z}^{j}z^{k}d\mu_{n},$Then there cists a Borel measure$e$$\mu_{0}$$n\in N_{0},$$j,$ $k\in N_{0}$supported on compact set$\langle T^{j}T^{k}u_{0},v_{0}\rangle=\int_{\sigma(T)}\overline{z}^{j}z^{k}d\mu 0$.

$n\in N_{0},$.

$\sigma(T)\subset \mathbb{C}$$j,$ $k\in N_{0}$such that.

3. Remarks and Problems

We close this note an open problem and related some remarks. Recall from [FJKP]
that if an operator of the form $T=N+K$ , where $N$ is a normal operator and $K$ is acompact operator, then has a moment sequence in version of [FJKP]. But we do not$T$know the following.

Problem 3.1. Let $T=N+K$ , where $N$ is a normal operator and $K$ is a compact?
-moment sequence supported onoperator. Does have
beLet $K:=K(\mathcal{H})$ be the set of compact operators on . Let :the Calkin map. Problem 3.1 is so interesting because of the following remark.
Remark 3.2. Assume that Problem 3.1 is true. If an operator $T$ is essentially normalor has a-moment sequence supported on(i.e., $\pi(T)$ is normal), then has$T$$a^{*}$$\sigma(T)$$\mathcal{H}$$T$$a^{*}$$\pi$$\mathcal{L}(\mathcal{H})arrow \mathcal{L}(\mathcal{H})/K$$\sigma(T)\subset \mathbb{C}$

$T$

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nontrivial invariant subspace. (Indeed, by BDF-theorem ([BDF]), if is biquasitriangular,. If $T$ is not biquasitriangular, by
has $a*$ -moment sequence supported onAFV-theorem ([AFV]), has a nontrivial invariant subspace.)
$T$

$T$

$\sigma(T)\subset \mathbb{C}$$T$

Remark 3.3. An operator
is called almost hyponomal if T’T–TT’ can$P\geq0$
be written as $P+K$ , where, the ideal of trace-class operators inand,(cf. [V]). It follows from [V] that if
is almost hyponormal,, then $T+X$ has ais the Hilbert-Schmidt class, and T’T–TT*
whereof the form $T+X$ , where
nontrivial invariant subspace. Thus, if every operator in$T$sequenceor has a nontrivial
is almost hyponormal andadmits-moment$T+X$ has a nontrivial’ invariant
invariant subspace. (Indeed, if $T$ “T–TT$T+X$ is essentially normal. By Remark 3.2, $T+X$ hassubspace. If T’T-TT’-moment sequence or $T+X$ has a nontrivial invariant subspace.)
The following is an open problem which is more general 3.1.
Problem 3.4. Does every essential normal operator have *-moment sequence ?
$T\in \mathcal{L}(\mathcal{H})$$K\in C_{1}(\mathcal{H})$$X\in C_{2}(\mathcal{H})$$T\in \mathcal{L}(\mathcal{H})$$\mathcal{L}(\mathcal{H})$$\not\in C_{1}(\mathcal{H})$$C_{2}(\mathcal{H})$$\mathcal{L}(\mathcal{H})$$X\in C_{2}(\mathcal{H})$$a^{*}$$”\not\in C_{1}(\mathcal{H}),$$\in C_{1}(\mathcal{H}),$$a^{*}$

Finally, we close this note with the following open problem.

Problem 3.5. Let be an invertible operator inDoes
have *-moment sequence ?
$T$$\mathcal{L}(\mathcal{H})$with$a^{*}$-moment sequence.

$T^{-1}$

References[AFV]C. Apostol, C. Foias and D. Voiculescu, Some results on non-quasitriangular
operators, IV, Revue Roum. de Math. Pure. Appl. 18 (1973), 487-514.

[AG]

A. Atzmon and G. Godefroy, An application of the smooth variational principle
to the eristence of nontrivial invariant subspaces, Comp. R. I’Acad. Sci. Paris,
Serie I, Math. 332(2001), 151-156.

[BDF]

L. Brown, R. G. Douglas and P. Fillmore, Extensions of C’-algebras and Khomology, Ann. Math. 105 (1977), 265-324.

[FJKP] C. Foias, I. Jung, E. Ko and C. Pearcy, Operators that admit a moment sequence,Israel J. Math. 145 (2005), 83-91.

[CJKP] B. Chevreau, I. Jung, E. Ko and C. Pearcy, Operators that admit a moment
sequence, Proc. the Amer. Math. Soc., 135 (2007), 1763-1767.
$\Pi$

[V]

D. Voiculescu, A note on $quasitri\cdot angularity$ and trace-class self-commutators,
Acta Sci. Math. (Sz.) 42 (1980), 1303-1320.