数理解析研究所講究録1080 巻 1999 年 1-5

1Operators: their Aluthge transforms and invariant$\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}_{1}\supset \mathrm{a}\mathrm{C}\mathrm{e}\mathrm{s}$Il Bong Jung, Eungil Ko, and Carl Pearcy
This paper will be appeared in other journal. Let be a separable, infinite dimensional, complex Hilbert space, and let
denote the algebra of all boundedlinear operators on . An arbitrary operator in
has a unique polar decomposition $T=UP$ , where $P=(T^{*}T)^{\frac{1}{2}}=|T|$ and is a partial isometry with initialspace the closure of the range ofand final space the closure of the range of .
Associated with there is a related operator, sometimes called
the Aluthge transform of because it was studied in [1] in the context that is ap-hyponormal operator (to be defined below). In this note we derive some spectralconnections between an arbitrary
and its associated Aluthge transformthat enable us, in particular, to generalize an-theorem ofBerger [2] to that context. We will also show that the hyperinvariant subspace
problems for hyponormal and p–hyponormal operators are equivalent.
The following lemma is completely elementary, but sets forth basic relations
between and that will be useful throughout the paper.
$\mathcal{H}$$\mathcal{L}(\mathcal{H})$$T$$\mathcal{H}$$\mathcal{L}(\mathcal{H})$

$U$

$T$

$|T|$

$T$

$\tilde{T}=|T|^{\frac{1}{2}}U|\tau|^{\frac{1}{2}}$

$T$

$T$

$T\in \mathcal{L}(\mathcal{H})$$\tilde{T}$$\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{t}_{-\mathrm{S}}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}$$T$$\tilde{T}$Lemma 1.1. Let $T=U|T|$ (polar decomposition) be an arbitrary operator inand let
be its Aluthge transform. Then$\mathcal{L}(\mathcal{H})$$\overline{T}=|T|^{\frac{1}{2}}U|\tau|^{\frac{1}{2}}$(1)$|T|^{\frac{1}{2}}\tau=\tilde{\tau}|T|^{\frac{1}{2}}$,and(2)$T(U|T|^{\frac{1}{2})}=(U|T|^{\frac{1}{2})\tilde{T}}$.

In $particular_{\mathrm{z}}T$ is a quasiaffinity (is one-to-one and has dense range) if andonly $if|T|$ is a quasiaffinity and is a unitary operator, so is a quasioffinity if
is. $Moreover_{y}$ in this $case_{;}T$ and are quasisimilar. Furthermore,? nvertible
and in this case,and are similar.
if and only if$i.e_{\mathrm{Z}}.T$

$U$

$T$

$\tilde{T}$$\tilde{T}$$T\dot{i}\circ$$T$$\tilde{T}is_{f}$$\tilde{T}$Remark 1.2. Consider the Hilbert space $\mathcal{H}=L^{2}([0,1], \mu)$ , where is Lebesgue
measure, and let
be any orthonormal basis foris thesuch that
$Ue_{n}=e_{n+1},$constant function 1. Let, so
be defined byisa unilateral shift, and consider $T=U(M_{x})^{2}$ , whereis multiplication by theposition function. Then is clearly not a quasiaffinity, but an easy calculation$\mu$$\{e_{n}\}_{n=1}^{\infty}$$\mathcal{H}$$e_{1}$$n\in \mathrm{N}$$U\in \mathcal{L}(\mathcal{H})$$M_{x}$

$T$

$U$

2
shows that $\tilde{T}=M_{x}UM_{x}$ is a quasiaffinity. Thus the corresponding implication inLemma 1.1 only goes one way.
Our first theorem shows that there is an intimate spectral connection between,
and its associated . As usual, we write(an arbitrary operator)
ofpoint spectrum
for the spectrum, point
and’ and approximate, respectively.
$\sigma(T),$$\overline{T}$$T$$\sigma_{ap}(T)$$\sigma_{p}(T)$$\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{C}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{m}.$$T$

Theorem 1.3. For every$\sigma_{p}(\tilde{T}),$

$T$

in

$\mathcal{L}(\mathcal{H}),$$\sigma_{ap}(T^{*})\backslash (0)=\sigma_{ap}((\tilde{T})*)\backslash (0)$, and$\sigma(T)=\sigma(\tilde{T}),$$\sigma ap(_{\vee}\tau)=\sigma_{a}(\tilde{\tau})_{r}\sigma_{p}(pT)=$$\sigma_{p}(T^{*})\backslash (0)=\sigma_{p}((\tau)*)\backslash (0)$.

We remark here that the example given in Remark 1.2 shows that all thespectral equalities in Theorem 1.3 are best possible.
for, and, we write, as usual,For an operator
respectively.
the essential (Calkin), left essential, and right essential spectra of ,in
if and only if there exists an orthonormal sequence
Recall thatsuch that $\lim_{narrow\infty}||(A-\lambda)e_{n}||=0$ , or, equivalently, if and only if there exists
converges weakly to zero andsuch thatof unit vectors ina sequence$||(A-\lambda)f_{n}||=0$ .
.
.
with associated Aluthge transform , weCorollary 1.4. For any.
, andhave$\sigma_{e}(A),$$A\in \mathcal{L}(\mathcal{H})$$\sigma_{re}(A)$$\sigma_{l\mathrm{e}}(A)$$A$$\{e_{n}\}$$\lambda\in\sigma_{le}(A)$$\mathcal{H}$$\{f_{n}\}$$\mathcal{H}$$\{f_{n}\}$$\lim_{narrow\infty}$$\tilde{T}$$T\in \mathcal{L}(\mathcal{H})$$\sigma\iota_{e}(T)=\sigma_{l}e(\tilde{T})$$\sigma_{e}(T)=\sigma_{e}(\tilde{T}),$$\sigma_{re}(T)\backslash (0)=\sigma re(\tilde{\tau})\backslash (\mathrm{o})$We turn now to the intimate connection between the invariant subspace latticesof an arbitrary operator and its associated . As usual, we write Lat $(A)$ for theis not. If
invariant subspace lattice of an arbitrary operator, so trivially has a nontrivial invariant
a quasiaffinity, then, itandrelation between
thesubspace. Thus when investigating
suffices to consider the case that is a quasiaffinity.
The following is an improvement of [5, Theorem 2].
$\tilde{T}$$T$$T\in \mathcal{L}(\mathcal{H})$$A\in \mathcal{L}(\mathcal{H})$$T$$0\in\sigma_{p}(T)\cup\sigma_{p}(T^{*})$$\mathrm{L}\mathrm{a}\mathrm{t}(T)$$\mathrm{L}\mathrm{a}\mathrm{t}(\tilde{T})$$T$

Theorem 1.5. Let $T=U|T|$ (polar decomposition) be an arbitrary quasiaffin. Then the mapping
ity in$\mathcal{L}(\mathcal{H})$$\phi$

maps$\mathrm{L}\mathrm{a}\mathrm{t}(T)$into Lat:$Narrow(|T|^{\frac{1}{2}\wedge^{\subset})^{-}}, N\in \mathrm{L}\mathrm{a}\mathrm{t}(T)$$(\tilde{T}))$and moreover if (0)(0)$\neq\lambda^{\subset}\neq \mathcal{H}$$\neq\phi(N)=(|\tau|\frac{1}{2}\Lambda^{\subset})-\neq \mathcal{H}$,, then.

Moreover the mapping$\psi$

maps Lat$(\tilde{T})$:$\mathcal{M}arrow(U|T|^{\frac{1}{2}}\mathcal{M})^{-}$

into Lat $(T)$ , and if(0),$(\mathrm{O})\neq M\neq \mathcal{H}$$M\in \mathrm{L}\mathrm{a}\mathrm{t}(\tilde{T})$, then$\neq\psi(\mathcal{M})=(U|\tau|\frac{1}{2}\mathcal{M})^{-}\neq \mathcal{H}$.

,3Consequently,$\mathrm{L}\mathrm{a}\mathrm{t}(T)$is nontrivial if and only if$\mathrm{L}\mathrm{a}\mathrm{t}(\tilde{T})$is nontrivial.

Remark 1.6. If in Theorem 1.5 is invertible, then and are similar (seeLemma 1.1), and thus have isomorphic invariant subspace lattices. Whether thisis true for an arbitrary noninvertible quasiaffinity , the authors have not beenable to determine. Note that Theorem 1.5 implies that if one is trying to solve the, it suffices toinvariant subspace problem for a particular quasiaffinity.
show that has a nontrivial invariant subspace.
$T$

$T$

$\overline{T}$$T$$T\in \mathcal{L}(\mathcal{H})$$\tilde{T}$and satisfies $(T^{*}T)^{p}\geq(TT^{*})^{p}$ forDefinition 1.7 $[1,6]$ . Suppose
is called a -hyponormal operator. If
some in the interval $(0, +\infty)$ . Thenis sometimes called semi-hyponormal [1] and if $p=1,$ is hyponormal.
$T\in \mathcal{L}(\mathcal{H})$

$T$

$p$

$p= \frac{1}{2},$

$p$

$T$

$T$

There is a vast literature concerning p–hyponormal operators (for $0<p<$1) in which various special cases of Theorem 1.3 are proved (for rhyponormal
uences of L\"owner’s
cothepurposes, we needoperators).

For.our.

$0.\mathrm{n}1\mathrm{y}$$\mathrm{f}\mathrm{o}\mathrm{l}1_{0}..\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}_{1}$$.$$\backslash$nse..q$\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}.[8]$is a p-hyponorin operator for some in the
Remark 1.8. If
interval $(0, +\infty)$ , then is also -hyponormal for every $0<q\leq p$ . In particular,
an operator that is a p-.hyponormal operator for some $p>1$ is also hyponormal.
Thus the interest in p–hyponormal operators has been concentrated mainly (butponormal operators for which; cf., for example, [4] on
not’$\mathrm{a}1$$T\in \mathcal{L}(\mathcal{H})$

$T$

$p$

$q$$)$$\mathrm{t}\mathrm{h}\mathrm{o}.\mathrm{s}\mathrm{e}p- \mathrm{h}\backslash .\mathrm{y}$$\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{i}_{\mathrm{V}}\mathrm{e}11^{\cdot}.\mathrm{y}$$0<p\leq 1$ .

The following lemma was proved in [1] in the special case in which the partial
isometry in the polar decomposition $T=U|T|$ is a unitary operator, but theproof carries over to the general case. We introduce the proof here for conveniences.
$U$Lemma $1.9([1])$ . Suppose that $T=U|T|$ (polar decomposition) is an. Then its Aluthge transform-hyponormal operator infor someoperator.
hyponormalais$arbitrar\underline{y}$$\mathcal{L}(\mathcal{H})$$p$$p \in[\frac{1}{2},1]$

was also proved in [1] in case $T=U|T|$ with
The following
operator, but once again, the proof can be made to work in general.
$\dot{1}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a}$$U$$T$a unitary

Lemma $1.10([1])$ . Suppose $T=U|T|$ (polar decomposition) is an arbitrary
is a. Then-hyponormal operator infor some in the interval-hyponormal operator and the Aluthge $trans.formT\approx of$ is a hyponormal
operator.
$\mathcal{L}(\mathcal{H})$$p$$(0, \frac{1}{2})$$p$$\tilde{T}$$\tilde{T}$$(p+ \frac{1}{2})$If is a bounded open set in the complex plane , recall that a subset A
is said to be dominating for if every bounded function $h(z)$ holomorphic onsatisfies$\mathrm{C}$$\mathcal{U}$$\mathcal{U}$$\sup_{\mathcal{Z}\in u}|h(z)|=z\in \mathcal{U}\sup_{\Lambda\cap}|h(z)|$First we recapture the following corollary.

.

$\subset \mathcal{U}$$\mathcal{U}$4Corollary 1.11[5, Theorem 3] Suppose $T=UP$ (polar decomposition) is
an arbitrary -hyponormal operator for someand suppose that thereexists a nonempty open set insuch thatis dominating for . Then
has a nontrivial invariant subspace.
$p\in(\mathrm{O}, +\infty)_{J}$$p$$\mathrm{C}$$\mathcal{U}$$\sigma(T)\cap \mathcal{U}$$\mathcal{U}$$T$

The following theorem generalizes a surprising theorem of Berger [2] for hyponormal operators to the context of p–hyponormal operators.
Theorem 1.12. Let
be an arbitrary -hyponormal operator for. Then there exists a positive integer such that for all positivesomeintegers $k\geq K,$
has a nontrivial invariant subspace.
$T\in \mathcal{L}(\mathcal{H})$$p$$p\in(\mathrm{O}, +\infty)$$I\mathrm{t}’$$T^{k}$

of is a nontrivial hyperinvariant subspace for an
Recall that a subspaceoperator
if (0)
andis invariant under every operator in$\{S\in\mathcal{L}(\mathcal{H}):
ST=TS\}$
the commutantof . We write Hlat $(A)$ for the lattice ofhyperinvariant subspaces of an operator. If
and is nota quasiaffinity, thenand Hlat
for trivial reasons.
$(T)$
Then when investigating the relation between Hlat, it suffices toand Hlat
that the case in which is a quasiaffinity.
$\mathcal{M}$$\mathcal{H}$$\neq M\neq \mathcal{H}$$T\in \mathcal{L}(\mathcal{H})$$\mathcal{M}$$T$$0\neq T\in \mathcal{L}(\mathcal{H})$$A\in \mathcal{L}(\mathcal{H})$$T$$(T)\neq\{(0), \mathcal{H}\}$$0\in\sigma_{p}(T)\cup\sigma_{p}(\tau*)$$(\tilde{T})$$T$

Theorem 1.13 Let
be an arbitrary nonzero quasiaffinity. Thenhas a nontrivial hyperinvariant subspace if and only if its Aluthge transformdoes. Thus the hyperinvariant subspace problem for -hyponormal operators (any$p\in(0, +\infty))$ is equivalent to the hyperinvariant subspace problem
for hyponormal
operators.
$T\in \mathcal{L}(\mathcal{H})$$T$$\overline{T}$$p$

Recall thatis ais invertible and-hyponormal operator if$\log(\tau\tau*)\leq\log(\tau*\tau)$ .
Note that any invertible p–hyponormal operator is loghyponormal.
$T$$\log$$T\in \mathcal{L}(\mathcal{H})$

Theorem 1.14 Suppose $T=UP$ (polar decomposition) is an arbitrary loghyponormal operator, and suppose that there exists a nonempty open set
insuch thatis dominating. Then has a nontrivial invariant subspace.
$\mathcal{U}$$T$$for\mathcal{U}$$\sigma(T)\cap \mathcal{U}$$\mathrm{C}$Recall thatis an -hyponormal operator if is -hyponormal for
any natural number . Note that any -hyponormal operator is p-hyponormal
for any positive real number .
We now close the paper as the following problem.
$T\in \mathcal{L}(\mathcal{H})$$T$$\infty$

$n$

$n$

$\infty$$p$

Problem 1.15 Suppose
is an arbitrary
have a nontrivial invariant subspace.
$T$$\infty$-hyponormal operator. Does$T$

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A. Aluthge, On -hyponormal operators for 0 $<p<$ 1, Integral Equations and Operator Theory 13(1990), 307-315.
$p$

5
[2]

C. Berger, Sufficiently high powers of hyponormal operators have rationally invariant subspaces, Integral Equations and Operator Theory
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[3]

S. Brown, Hyponormal operators with thick spectrum have invariant subspaces, Ann. of Math. 125(1987), 93-103.

[4]

R. Curto, P. Muhly and D. Xia, A trace estimate for p-hyponormal
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[5]

B. Duggal, -hyponormal operators and invariant subspaces, Acta Sci.
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[6]

M. Fujii and Y. Nakatsu, On subclasses of hyponorma\prime l operators, Proc.
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[7]

T. Hoover, Quasi-similarity of perators, Illinois J. Math. 16(1972), 678-$p$$o$686.
[8]K. L\"owner, Uber monotone

514.

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[9]M. Martin and M. Putinar, Lectures on hyponormal operators,
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Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space,
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Department of Mathematics
Kyungbook National University
Taegu 702-701, Korea
andDepartment of Mathematics
Ewha Women’s University
Seou1120-750, Korea
andDepartme-nt of Mathematics
Texas A&M University
College Station, TX 77843, U.S.A.