FURUTA INEQUALITY AND $p$-$\omega A(s,t)$ OPERATORS (The research of geometric structures in quantum information based on Operator Theory and related topics)

FURUTA

_INEQUALITY

AND

_{p-wA(s, t)}

OPERATORS

M.

\mathrm{C}\mathrm{H}\overline{\mathrm{O}}

, T.

PRASAD,

M.

RASHID,

K. TANAHASHI AND

A. UCHIYAMA

Dedicatedtothe memory ofProfessor

_Takayuki

Furuta with

_deep

_gratitude

ABSTRACT. The aim ofthis paper is to introduce small

_history

with Furata’s

_inequality

and

_relating

class of

_p‐wA

_{(s, t)}

_operators.

1. INTRODUCTION

Let

_{B(\mathcal{H})}

be the

_algebra

of all bounded linear

_operators

on a

complex

Hilbert _space\mathcal{H}. In

1987,

Furuta

[5]

proved

the

follwing inequality.

Theorem 1.

_[Furuta

_inequality]

Let 0 _<p,q,r \in \mathbb{R} and

_A,

B \in

B(\mathcal{H})

_satisfy

0

_\leq

B

_{\leq A.}

_{If p+2r}

_\leq

(1+2r)q

and

1\leq q

, then

B\displaystyle \frac{p+2r}{q}

\leq(B^{r}A^{p}B^{ $\Gamma$})^{\frac{1}{q}}

and

(A^{r}B^{p}A^{r})^{\frac{1}{\mathrm{q}}}\displaystyle \leq A\frac{p+2r}{\mathrm{q}}

This is a

good

extensionof Löwner‐Heinz’s

inequality

([7]

and

[12]).

Theorem 2.

_{[Löwner‐Heinz’s inequality]}

Let

A,

B\in B(\mathcal{H})

satisfy

0\leq B\leq A

and

0<p\leq 1

. Then

B^{p}\leq A^{p}.

Recall that an

_operator

T is saidto be

_hyponormal

if

T^{*}T\geq TT^{*}

. For

T\in B(\mathcal{H})

,set

|T|=(T^{*}T)^{\frac{1}{2}}

asusual.

By taking

U|T|x=Tx

for x\in \mathcal{H}and

Ux=0 for

_{x\in \mathrm{k}\mathrm{e}\mathrm{r}|T|,}

T hasa

_unique

polar decomposition

T=U|T|

with

\mathrm{k}\mathrm{e}\mathrm{r}U=\mathrm{k}\mathrm{e}\mathrm{r}|T|

.

Aluthge

[1]

defined

Aluthge

transformation

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

and studied

_{interesting properties}

of

_{p‐‐hyponormal}

_operators

for

_{0<p\leq 1.}

Definition 3. T \in

B(\mathcal{H})

is said to be

_{p‐‐hyponormal}

if

_(T^{*}T)^{p}

\geq

(TT^{*})^{p}

where

_p\in(0,1].

The class of r

‐hyponormal

operators

is a

generalization

ofthe class of

hyponormal

operators

by

Löwner‐Heinz’s

_inequality.

_Aluthge

_[1]

_proved

follwing

result.

Theorem 4. Let T bep

‐hyponormal. If

0<p\leq 1/2

, then

\tilde{T}

is

(p+1/2)-hyponormal.

If

1/2\leq p\leq 1

, then

\tilde{T}

is

hyponormal.

This is a _epoc

making

result.

Aluthge

transformation is a

_strong

tool

of

_operator

_theory

_{and many}

_applications

has been

_studied,

for

_example,

Putnam

_ineqality,

_Fuglede

Putnam

_type

_{theorem, Wyle}

_type

theorem. I

think that

_{generalization}

ofclassof

_operators

may be

good

wayto

_investigate

non‐normal

_operators.

Furuta

_[6]

and Yoshino

_[17]

definded

_generalized

transformation

_{T(s, t)=}

_{|T|^{S}U|T|^{t}}

with

_0<s,

t and

_Yanagida

_[15],

Ito

_[8],

Definition 5. T is saidto be

_{wA(s, t)}

if

(1.1)

(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{t}{s+t}} \geq|T^{*}|^{2t}

and

(1.2)

|T|^{2s}\geq(|T|^{s}|T^{*}|^{2t}|T|^{s})^{\frac{s}{s+t}}.

Hence

_{generalized Aluthge}

transformation

_T(s,t)

of

_{wA(s, t)}

_operator

T

enjoys

the

_following

_property.

Proposition

6. Let T be

_{wA(s, t)}

. Then

|T(s, t)|^{\frac{2t}{s+t}} \geq|T|^{2t}

and

|T|^{2s}\geq|T(s, t)^{*}|^{\frac{2 $\epsilon$}{s+t}}

Hence

|T(s,t)|^{\frac{2r}{s+t}} \geq|T|^{2r}\geq|T(s,t)^{*}|^{\frac{2r}{s+t}}

for

all

_r\in(0,

_{\displaystyle \min\{s,}

t

Ito and Yamazaki

_[9]

_proved

that

_(1.1)

_implies

_(1.2).

This is a

_good

result. Thismeansthat class of

wA(s,t)

_operators

arecoincideswith

A(s, t)

operators.

Definition 7. T is saidto be

_{A(s, t)}

if

(1.3)

(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{t}{s+t}}\geq|T^{*}|^{2t}.

Class

_A(1,1)

is said to be class A and class

_A(1/2,1/2)

is said to be

w

‐hyponormal

[4,

9,

15].

Prasad and Tanahshi

[16]

definded

p‐wA

(s, t)

_op‐

erator for

_{0<p\leq 1}

and

_{0<s, t, s+t\leq 1}

as follows.

Deflnition 8. T is saidto be

_p‐wA

_{(s, t)}

if

(1.4)

(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{\mathrm{p}\mathrm{t}}{ $\epsilon$+t}}\geq|T^{*}|^{2pt}

and

(1.5)

|T|^{2ps}\geq(|T|^{s}|T^{*}|^{2t}|T|^{s})^{\frac{\mathrm{p}s}{s+\mathrm{t}}}.

Hence Í\succwA

(s, t)

operator

is a

generalization

of

wA(s, t)

_operator

by

Löwner‐Heinz’s

_inequality.

The aim of_{this paper is} to prove several prop‐

erties of

_{p-wA(s, t)}

_operaor and show some _open

problems

of

p‐wA

(s, t)

operaor. Main results are

proved

in

[16]

and

[2].

2. RESULTS

At

_first,

weshow

generalized Aluthge

transformation

T(s,t)

of

p‐wA

(s,t)

operator

T

_enjoys

the

_following

_property

_[16].

Theorem 9. Let T be_p‐

_{wA(s, t)}

. Then

|T(s, t)|^{\frac{2pt}{s+t}} \geq|T|^{2pt}

and

|T|^{2ps}\geq|T(s, t)^{*}|^{\frac{2}{s}\mathrm{g}_{\frac{s}{t}}}+

Hence

for

all

_r\in(0,

_{\displaystyle \min\{s,}

t

Proof.

(|T^{*}|^{\mathrm{t}}|T|^{2s}|T^{*}|^{t})^{4_{\frac{t}{+t}}}\mathrm{s} \geq|T^{*}|^{2pt}

\Leftrightarrow(U|T|^{t}U^{*}|T|^{2s}U|T|^{t}U^{*})^{\frac{\mathrm{p}t}{ $\epsilon$+t}}\geq U|T|^{2pt}U^{*}

\Leftrightarrow U(|T|^{t}U^{*}|T|^{2s}U|T|^{t})^{\mathrm{g}_{\frac{\mathrm{t}}{t}}}\dot{s}+U^{*}\geq U|T|^{2pt}U^{*}

(

[8

, Lemma

2.1])

\Leftrightarrow(|T|^{t}U^{*}|T|^{2s}U|T|^{t})^{\frac{pt}{8+t}}

\geq|T|^{2pt}

( [8

, lemma

2.1])

\Leftrightarrow|T(s, t)|^{\frac{2_{ $\Gamma$)}t}{ $\epsilon$+t}} \geq|T|^{2pt}.

Also,

(|T|^{s}|T^{*}|^{2t}|T|^{s})^{\frac{ps}{ $\epsilon$+t}}\leq |T|^{2ps}

\displaystyle \Leftrightarrow(|T|^{s}U|T|^{2t}U^{*}|T|^{s})^{1}\ovalbox{\tt\small REJECT}\frac{8}{+t} \leq|T|^{2ps}

\Leftrightarrow|\{T(s, t)\}^{*}|^{\frac{2ps}{s+t}} \leq |T|^{2ps}.

\square

Nextwe show classof

p‐wA

(s, t)

_operators

are

decreasing

class of opera‐

torswith

_{0<p\leq 1}

and

_increasing

with

_0<s,

t\leq 1

. The

proof

is

essentially

due to C.

Yang

and J. Yuan

_([19]

_Proposition

_3.4).

Lemma 10.

_If

T is

_p‐wA

_{(s, t)}

and

0<s\leq s_{1}, 0<t\leq t_{1}, 0<p_{1} \leq p\leq 1,

thenT is

_{p_{1}-wA(s_{1}, t_{1})}

.

Proof.

Let Tbe

_p‐wA

_{(s, t)}

. Then

(2.1)

(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t^{t}})^{B}\dot{s}+\overline{t} \geq |T^{*}|^{2tp}

and

(2.2)

|T|^{2sp}\geq(|T|^{s}|T^{*}|2t^{s}|T|^{S})^{-B_{\overline{t}}}s+.

We prove that T is

p\leftrightarrow wA(s_{1}, t_{1})

. Then T is

p_{1}-wA(s_{1}, t_{1})

by

Lowner‐Heinz’s

inequality.

Let

A_{1}

=

(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{ip}{s+\mathrm{t}}}

and

B_{1}

=

|T^{*}|^{2tp}

. Since

(1)

imphes A_{1}

\geq

B_{1}

, wehave

(B^{\frac{r2}{1^{2}}}A_{1}^{p_{2}}B^{\frac{r_{2}}{1^{2}}})^{\frac{1}{p_{2}}}+r\vec{+r_{2}} \geq B_{1}^{1+r2}

for any

r_{2}>0

and

_{p_{2}\geq 1}

_by

Furuta’s

_inequality

_[5].

Let

$\beta$\displaystyle \geq t,p_{2}=\frac{s+t}{tp}\geq 1, r_{2}=\frac{ $\beta$-t}{tp}\geq 0.

Then

(|T^{*}|^{ $\beta$}|T|^{2s}|T^{*}|^{ $\beta$})^{\frac{tp+ $\beta$-t}{s+ $\beta$}} \geq|T^{*}|^{2tp+2 $\beta$-2t}.

Hence wehave

(|T^{*}|^{ $\beta$}|T|^{2s}|T^{*}|^{ $\beta$})^{\frac{w}{s+ $\beta$}}\geq|T^{*}|^{2w}

Let

for

_{$\beta$\geq t}

. Then

f_{s}( $\beta$)= (|T|^{s}|T^{*}|^{2 $\beta$}|T|^{s})^{\frac{s}{s+ $\beta$}}

f_{s}( $\beta$)=\{(|T|^{s}|T^{*}|^{2 $\beta$}|T|^{s})^{\frac{s+ $\beta$+w}{s+ $\beta$}}\}^{\frac{s}{s+ $\beta$+w}}

=\{|T|^{s}|T^{*}|^{ $\beta$}(|T^{*}|^{ $\beta$}|T|^{2s}|T^{*}|^{ $\beta$})^{\frac{w}{s+ $\beta$}}|T^{*}|^{ $\beta$}|T|^{s}\}^{\frac{s}{ $\epsilon$+ $\beta$+w}}

\geq\{|T|^{s}|T^{*}|^{ $\beta$}|T^{*}|^{2w}|T^{*}|^{ $\beta$}|T|^{s}\}^{\frac{s}{s+ $\beta$+w}}

=\{|T|^{s}|T^{*}|^{2( $\beta$+w)}|T|^{s}\}^{\frac{\mathrm{s}}{\'{o}+ $\beta$+w}}

=f_{s}( $\beta$+w)

.

Hence

_{f_{s}( $\beta$)}

is

_decreasing

for

_{$\beta$\geq t.}

Then, by

(2.2),

|T|^{2sp}\geq(|T|^{s}|T^{*}|^{2t}|T|^{s})^{\frac{\mathrm{s}p}{s+t}}

=\{f_{s}(t)\}^{p}

\geq\{f_{s}(t_{1})\}^{p}=(|T|^{s}|T^{*}|^{2t_{1}}|T|^{s})^{-}\ovalbox{\tt\small REJECT} +t_{1}sp

Let

_{A_{2}=|T|^{2sp}}

and

B_{2}=

(|T|^{s}|T^{*}|^{2t_{1}}|T|^{s})^{\frac{S\mathrm{f}^{\mathrm{j}}}{s+t_{1}}}

. Then

A_{2}^{1+r3}\geq (A_{2}\not\simeq^{r}B_{2}^{p_{3}^{r}}A_{2}^{\Rightarrow})^{\frac{1+r3}{\mathrm{p}_{3}+r_{3}}}

for any

r3\geq 0

and

_{p3\geq 1}

_by

Furuta’s

_inequality

_[5].

Let

p_{3}=\displaystyle \frac{s+t_{1}}{sp}\geq 1, r3=\frac{s_{1}-s}{sp}\geq 0.

Then

|T|^{2sp+2s_{1}-2s}\geq (|T|^{s_{1}}|T^{*}|^{2t_{1}}|T|^{s_{1}})^{\frac{ $\epsilon$ p+s_{1-B}}{s_{1}+t_{1}}}

Since

sp+s_{1}-s-s_{1}p=(s_{1}-s)(1-p) \geq 0,

wehave

|T|^{2s_{1}p}\geq (|T|^{s_{1}}|T^{*}|^{2t_{1}}|T|^{s_{1}})^{\frac{s_{1}p}{81+t_{1}}}

Similarly,

wehave

(|T^{*}|^{t_{1}}|T|^{2s_{1}}|T^{*}|^{t_{1}})^{\frac{t}{s_{1}}\mathrm{L}_{\overline{t_{1}}}^{p}}+ \geq|T^{*}|^{2t_{1}p}.

Hence T is

_p‐wA

_{(s_{1}, t_{1})}

. \square

The

_following

result seems_new, even for class

A(s, t)

operators.

Proof.

Let

_T=U|T|

the

_{polar decomposition}

of T. Then

|T^{-1}|^{2}=(T^{-1})^{*}T^{-1}=(T^{*})^{-1}T^{-1}=(TT^{*})^{-1}=|T^{*}|^{-2}.

Hence

|T^{-1}|=|T^{*}|.

Also,

|(T^{-1})^{*}|^{2}=(T^{-1})(T^{-1})^{*}=T^{-1}(T^{*})^{-1}=(T^{*}T)^{-1}=|T|^{-2}.

Hence

|(T^{-1})^{*}|=|T|^{-1}.

Then

\{|(T^{-1})^{*}|^{s}|T^{-1}|^{2t}|(T^{-1})^{*}|^{s}\}^{\frac{ $\epsilon$ \mathrm{p}}{s+l}}

=(|T|^{-s}|T^{*}|^{-2t}|T|^{-s})^{\frac{ $\epsilon$ p}{s+l}}

=(|T|^{s}|T^{*}|^{2t}|T|^{s})^{-p}\overline{ $\epsilon$}+^{\frac{s}{t}}

\geq|T|^{-2sp}=|(T^{-1})^{*}|^{2s\mathrm{p}}

and

\{|T^{-1}|^{t}|(T^{-1})^{*}|^{2s}|T^{-1}|^{t}|\}^{\frac{tp}{\^{o}+t}}

=\{|T^{*}|^{-t}|T|^{-2s}|T^{*}|-t^{\mathrm{t}}\}^{p_{\overline{l}}}8+

=(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{-tp}{s+t}}

\leq|T^{*}|^{-2tp}=|T^{-1}|^{2tp}.

\square

Corollary

12.

_If

T is

_{A(s, t)}

and T is

_invertible,

then

T^{-1}

_{\dot{u}A(t, s)}

.

Let 0 <p

\leq

1 and

_S,

T \in

_{B(\mathcal{H})}

be non zero

_operators.

In

[3],

Duggal

proved

that tensor

_product

T\otimes S is

_{p‐‐hyponormal}

ifand

_only

if Tand S are

p‐‐hyponormal.

The passage of calss A

_operators

isstudied

by

Jeon and

Duggal

[10].

Tanahashi and Chō

_[13]

_proved

that the tensor

_product

T\otimes S

is ofclass

_{A(s, t)}

if and

_only

if T andS are class

A(s, t)

. Nowwewillprove

similar_{result for p\rightarrow‐class}

_{wA(s, t)}

_operators

_{by adopting}

the ideas in

_[13],[10].

Lemma 13.

_[11]

Let

_{T_{1}, T_{2}, S_{1},}

S_{2}

\in

_{B(\mathcal{H})}

be non

negative operators.

If

T_{1}\neq 0

and

_{S_{1}\neq 0}

, then the

following

conditions are

equivalent.

(1)

T_{1}\otimes S_{1}\leq T_{2}\otimes S_{2}.

(2)

There existsc>0 such that

_{T_{1}\leq \mathrm{c}T_{1}}

and

_{S_{1}\leq c^{-1}S_{2}.}

Lemma 14.

_[13]

Let

_T=U|T|

and

_S=V|S|

be the

_polar

_{decompositions}

of

T,

S\in B(\mathcal{H})

. Then the

following

assertions hold.

(1) |T\otimes S|=|T|\otimes|S|.

(2)

T\otimes S=(U\otimes V)(|T|\otimes|S|)

is the

_{polar decomposition}

_of

T\otimes S.

(3)

(T\otimes S)(s,t)=T(s, t)\otimes S(s, t)

for

s,t>0.

Theorem 15. Let

_S,

T \in

B(\mathcal{H})

be non zero

_operators.

Then

T\otimes S

is

Proof.

Let

_S,

T \in

B(\mathcal{H})

be non zero

p‐‐class

wA(s, t)

_operators

and S =

V|S|,

T=

_U|T|

be

_{polar decomposions}

of

_S,

T. Then

|T(s, t)|^{\frac{2tp}{s+t}}

\geq

|T|^{2tp}

and

|S(s, t)^{*}|^{\frac{2t\mathrm{p}}{\mathrm{s}+t}}

\geq

_|S|^{2tp}

by

Theorem 9.

_{By applying}

Lemma

_14,

weobtain

|(T\otimes S)(s, t)|^{\frac{2tp}{s+l}}=|T(s, t)\otimes S(s, t)|^{\frac{2t\mathrm{p}}{s+i}}

=|T(s, t)|^{\frac{2tp}{s+t}}\otimes|S(s, t)|^{\frac{2tp}{s+t}}\geq |T|^{2t\mathrm{p}}\otimes|S|^{2tp}=|T\otimes S|^{2tp}.

Similarly,

wehave

|T\otimes S|^{2sp}\geq|\{(T\otimes S)(s, t)\}^{*}|^{\frac{2s}{\^{o}}R}+t.

HenceT\otimes S is

_p‐wA

_{(s, t)}

.

Conversely,

supposethat

T\otimes S

is

_p‐wA

_{(s, t)}

. Then

|(T\otimes S)(s, t)|^{\frac{2}{s}}+ts_{t}=|T(s, t)|^{\frac{2t\mathrm{p}}{s+t}}\otimes|S(s, t)|+

\geq|T|^{2tp}\otimes|S|^{2tp}=|T\otimes S|^{2tp}

and

|T\otimes S|^{2\mathrm{s}p}\geq|\{(T\otimes S)(s, t)\}^{*}|^{\frac{2s}{8}R}+t.

Hence there exists c>0 such that

c|T(s, t)|^{\frac{2}{8}}+t\mathrm{t}_{\mathrm{A}}\geq|T|^{2tp}

and

c-1^{A}|S(s, t)|^{\frac{2}{s}}+tt\geq|S|^{2tp}

by

Lemma 13. Let xbe aunit vector. Then

\Vert|T|^{tp}x\Vert^{2}=\langle|T|^{2tp}x, x\}\leq\langle c|T(s, t)|^{\frac{2\mathrm{t}\mathrm{p}}{s+t}}x, x\}

\leq c\Vert|T(s, t)|^{\frac{t}{8}}+^{R_{\overline{t}}}\Vert^{2}=c\Vert T(s, t)\Vert+tt

=c\Vert|T|^{s}U|T|^{t}\Vert^{\frac{2t\mathrm{p}}{s+l}}

\leq c(\Vert|T|^{s}\Vert\cdot 1\cdot\Vert|T|^{t}\Vert)^{\frac{2}{s}B^{t}}+t=c\Vert|T|^{tp}\Vert^{2}.

Hence

1\leq c

.

Similarly,

\Vert|S|^{tp}x\Vert^{2}=(|S|^{2tp}x, x\rangle\leq\langle c^{-1}|S(s, t)|^{\frac{2tp}{s+t}}x, x\rangle

\leq c^{-1}\Vert|S(s, t)|^{\frac{t\mathrm{p}}{s+t}}\Vert^{2}=c^{-1}\Vert S(s,t)\Vert^{\frac{2t\mathrm{p}}{s+i}}

=c^{-1}\Vert|S|^{s}V|S|^{t}\Vert^{\frac{2tp}{ $\epsilon$+t}}

\leq c^{-1}(\Vert|S|^{s}\Vert\cdot 1\cdot\Vert|S|^{t}\Vert)^{\frac{2i\mathrm{p}}{s+\mathrm{t}}}=\mathrm{c}^{-1}\Vert|S|^{tp}\Vert^{2}.

Hence

_{1\leq c^{-1}}

. Hence c=1. This

implies

|T(s, t)|^{\frac{2tp}{ $\epsilon$+t}}\geq|T|^{2tp}

and

|S(s, t)|^{\frac{2t}{s+}\mathrm{g}_{l}}\geq|S|^{2tp}.

Similarly

wehave

|T|^{2sp}\geq|\{T(s, t)\}^{*}|^{\frac{2sp}{s+\mathrm{t}}}

and

ThusT and S are

rwA(s, t)

. \square

Corollary

16. Let

_S,

T \in

_{B(\mathcal{H})}

be non zero

_operators.

Then

T\otimes S

is

p‐A

(s, t)

if

and

_{only if S,}

T are

p‐A

(s, t)

.

Theorem 17. Let T \in

_{B(\mathcal{H})}

be

_p‐wA

_{(s, t)}

with 0 < s,

t,

s+t = 1 and

0<p\leq

1. Let

$\rho$ e^{i $\theta$}\in \mathbb{C}

be an isolated

point

of

$\sigma$(T)

and

0< $\rho$

. Then the

Riesz

_idempotent

E

_for

T with

_respect

to

$\rho$ e^{i $\theta$}

is

_{self‐adjoint}

with

ran

E=\mathrm{k}\mathrm{e}\mathrm{r}(T- $\rho$ e^{i $\theta$})=\mathrm{k}\mathrm{e}\mathrm{r}((T- $\rho$ e^{i $\theta$})^{*})

.

and coincides with the Riesz

_idempotent

_{E(s, t)}

_for

_{T(s, t)}

with

_respect

to

i $\theta$

$\rho$ e .

Proof.

Since

_{$\sigma$(T)= $\sigma$(T(s, t))}

_by

Lemma 6 of

_[14],

_{$\rho$ e^{i $\theta$}}

is anisolated

_point

of

_{$\sigma$(T(s,}

t Since

_{T(s, t)}

is

_{rp‐hyponormal}

for all

_{r\displaystyle \in(0, \min\{s, t E(s, t)}

is

_{self‐adjoint}

and satisfies

ran

E(s, t)=\mathrm{k}\mathrm{e}\mathrm{r}(T(s, t)- $\rho$ e^{i $\theta$})

=\mathrm{k}\mathrm{e}\mathrm{r}(T- $\rho$ e^{i $\theta$})

and

$\rho$ e^{i $\theta$}\not\in $\sigma$(T(s, t)|_{\mathrm{r}\mathrm{a}\mathrm{n}E(s,t)})

.

Since

_{\mathrm{k}\mathrm{e}\mathrm{r}(T- $\rho$ e^{i $\theta$})=}

ran

E(s, t)

reduces T, wehave

T= $\rho$ e^{i $\theta$}\oplus T'

on \mathcal{H}=ran

E(s, t)\oplus

ran

(1-E(s,

t

Then T' is also class

_p‐wA

_{(s, t)}

and

_{T'(s, t)}

=

T(s, t)|_{\mathrm{r}\mathrm{a}\mathrm{n}}(1-E(s,t))

. Hence

$\rho$ e^{i $\theta$}\not\in $\sigma$(T'(s, t))= $\sigma$(T')

by

Lemma 6 of

_[14].

Hence

T'- $\rho$ e^{i $\theta$}

is invertible

and

T- $\rho$ e^{i $\theta$}=0\oplus(T'- $\rho$ e^{i $\theta$})

. This

implies

\mathrm{k}\mathrm{e}\mathrm{r}(T- $\rho$ e^{i $\theta$})=\mathrm{k}\mathrm{e}\mathrm{r}((T- $\rho$ e^{i $\theta$})^{*})

and

E=\displaystyle \frac{1}{2 $\pi$ i}\int_{ $\gamma$}(z- $\rho$ e^{i $\theta$})^{-1}\oplus(z-T')^{-1}dz=1\oplus 0=E(s, t)

where $\gamma$ isa small circle

containing

$\rho$ e^{i $\theta$}.

\square

Theorem 18. Let

_{T\in B(\mathcal{H})}

be

_p‐wA

_{(s, t)}

with

_0<s,

_{t, s+t\leq 1}

and 0<

p\leq 1

. Let

(T- $\rho$ e^{i $\theta$})x_{n}\rightarrow 0

for

x_{n}\in \mathcal{H}

with

\Vert x_{n}\Vert=1

and

$\rho$ e^{i $\theta$}\in \mathbb{C},

0< $\rho$.

Then

_{(|T|- $\rho$)x_{n},}

_{(U-e^{i $\theta$})x_{n}, (U-e^{i $\theta$})^{*}x_{n}, (T- $\rho$ e^{i $\theta$})^{*}x_{n}\rightarrow 0.}

Proof.

We may assume s+t=1

by

Lemma 10. Since

(T(s, t)- $\rho$ e^{i $\theta$})|T|^{s}x_{n}=|T|^{s}(T- $\rho$ e^{i $\theta$})x_{n}\rightarrow 0,

wehave

(T(s, t)- $\rho$ e^{i $\theta$})^{*}|T|^{s}x_{n}\rightarrow 0,

because

_{T(s, t)}

is

_{rp‐hyponormal}

for all

_{r\in(0, \displaystyle \min\{s,}

t Hence

0\leftarrow(T(s, t)- $\rho$ e^{i $\theta$})^{*}(T(s, t)- $\rho$ \mathrm{e}^{i $\theta$})|T|^{s}x_{n}

= (|T(s, t)|^{2}-$\rho$^{2})|T|^{s}x_{n}

- $\rho$ e^{-i $\theta$}(T(s, t)- $\rho$ e^{i $\theta$})|T|^{s}x_{n}- $\rho$(T(s, t)- $\rho$ e^{i $\theta$})^{*}|T' x_{n}.

This

_implies

and

Similarly,

wehave

(|T(s, t)|^{rp}-$\rho$^{rp})|T|^{s}x_{n}\rightarrow 0.

(|T(s, t)^{*}|^{rp}-$\rho$^{rp})|T|^{s}x_{n}\rightarrow 0.

Hence

0\leftarrow\langle(|T(s, t)|^{rp}-$\rho$^{rp})|T|^{s}x_{n}, |T|^{s}x_{n}\rangle

\geq\{(|T|^{rp}-$\rho$^{rp})|T|^{s}x_{n}, |T|^{s}x_{n}\}

\geq\langle(|T(s, t)^{*}|^{rp}-$\rho$^{r\mathrm{p}})|T|^{s}x_{n}, |T|^{s}x_{n}\rangle\rightarrow 0.

This

_implies

((|T|^{rp}-$\rho$^{rp})|T|^{s}x_{n}, |T|^{s}x_{n}\rangle\rightarrow 0.

Since

_{\displaystyle \frac{r}{2}\in(0,}

_{\displaystyle \min\{s,}

t we have

\langle(|T|\not\in^{r}-$\rho$^{r}\#)|T|^{s}x_{n}, |T|^{s}x_{n}\rangle\rightarrow 0

by

thesame

_argument.

Then

\Vert(|T|^{\frac{rp}{2}}-$\rho$^{\frac{rp}{2}})|T|^{s}x_{n}\Vert^{2}

=\langle(|T|^{\frac{r\mathrm{p}}{2}}-$\rho$^{B}r_{2})^{2}|T|^{s}x_{n},

|T|^{s}x_{n}\rangle

=((|T|^{rp}-$\rho$^{rp})|T|^{s}x_{n}, |T|^{s}x_{n}\rangle-2 $\rho$

ỉ

\yen

((|T|^{\frac{r\mathrm{p}}{2}}-$\rho$^{\frac{rp}{2}})|T|^{s}x_{n},

|T|^{s}x_{n}\rangle

\rightarrow 0. Hence

(|T|^{\frac{rp}{2}}-$\rho$^{\frac{r\mathrm{p}}{2}})|T|^{s}x_{n}\rightarrow 0

and so

(|T|- $\rho$)|T|^{s}x_{n}\rightarrow 0.

Then

|T|(|T|- $\rho$)x_{n}=|T|^{1-s}(|T|- $\rho$)|T|^{s}x_{n}\rightarrow 0.

Since

0=\displaystyle \lim\langle|T|(|T|- $\rho$)x_{n}, x_{n})

=\displaystyle \lim\Vert|T|x_{n}\Vert^{2}- $\rho$\lim\langle|T|x_{n}, x_{n}\rangle

=\displaystyle \lim\Vert Tx_{n}\Vert^{2}- $\rho$\lim(|T|x_{n}, x_{n}\}

=$\rho$^{2}- $\rho$\displaystyle \lim\langle|T|x_{n}, x_{n}\rangle,

we have

\langle|T|x_{n}, x_{n}\rangle\rightarrow $\rho$

and

\Vert(|T|- $\rho$)x_{n}\Vert^{2}=\Vert|T|x_{n}\Vert^{2}-2 $\rho$(|T|x_{n}, x_{n}\rangle+$\rho$^{2}

\rightarrow$\rho$^{2}-2$\rho$^{2}+$\rho$^{2}=0.

This

_implies

(|T|- $\rho$)x_{n}\rightarrow 0.

Since

and

_{0< $\rho$}

, wehave

Also,

(U-e^{i $\theta$})x_{n}\rightarrow 0.

\Vert(U-e^{i $\theta$})^{*}x_{n}\Vert^{2}=\Vert U^{*}x_{n}\Vert^{2}-\langle U^{*}x_{n}, e^{-i $\theta$}x_{n}\}-\{e^{-i $\theta$}x_{n}, U^{*}x_{n}\}+1

\leq 1-e^{i $\theta$}(x_{n}, Ux_{n})-e^{-i $\theta$}\langle Ux_{n}, x_{n}\rangle+1

\leq-e^{i $\theta$}\langle x_{n}, (U-e^{i $\theta$})x_{n}\rangle-e^{-i $\theta$}\{(U-e^{i $\theta$})x_{n}, x_{n}\rangle\rightarrow 0.

Hence

(U-e^{i $\theta$})^{*}x_{n}\rightarrow 0

and

(T- $\rho$ e^{i $\theta$})^{*}x_{n}=|T|(U-e^{i $\theta$})^{*}x_{n}+e^{-i $\theta$}(|T|- $\rho$)x_{n}\rightarrow 0.

\square

Open problems

It is known thatclass A

_operator

satisfiesPutnam

_type

_inequahty.

How‐

ever it is not known that Putnam

_type

_inequality

holds for

_p‐wA

_{(s, t)}

_op‐

erators. It seems a difficult

problem.

We note some _open

_problems

for

p‐wA

(s, t)

operators.

(1)

M. Ito and T. Yamazaki

_[9]

_proved

that

_{A(s, t)}

_implies

_{wA(s, t)}

. How‐

ever it isnot knownwhether

_p‐‐class

_{A(s, t)}

_{implies p‐wA}

_{(s, t)}

for

_0<p<1

ornot.

(2)

It is known that if T is class

_{A(s, t)}

and \mathcal{M} \subset \mathcal{H} is a T‐invariant

subspace,

then

_{T|_{\mathcal{M}}}

is class

_{A(s, t)}

. However it is not knownwhether this

property

holds for

_p‐wA

_{(s, t)}

_operator

T.

(3)

It is known that classA

_operator

T is normaloid. But it isnotknown

that

_p‐wA

_{(s, t)}

_operator

T is normaloidor not.

REFERENCES

[1]

A.

_Aluthge,

On_p

_{‐hyponormal}

_operators for 0 <p< 1 ,

Integral Equations

Operator Theory.

13

_(1990),

307‐ 315.

[2]

M.

_Cho‐,

M.

_Rashid,

K. Tanahashi and A.

_{Uchiyama, Spectrum of}

class_p‐

wA(s, t)

operators, Acta Sci. Math.

_(Szeged),

82

_(2016),

641‐649.

[3]

B.P

_Duggal,

TensorProduct of_{operators‐strong}

_stability

and\mathrm{p}

‐hyponormality,

Glasgow

Math. J

_{42.(200),371-381.}

[4]

M.

_Fujii,

D.

_Jung,

S. H.

_Lee.,

M. Y.

_Lee.,

and R.

_Nakamoto,

Some classes

of

operators relatedto

_paranormal

and

_{log hyponormal}

operators, Math.

_Japon.,

51

_(2000),

395A02.

[5]

T.

_Furuta,

A_{\geq B\geq 0} assures

(B^{r}A^{p}B^{r})^{\frac{1}{q}}

\geq B^{\leftarrow+\underline{2r}}\mathrm{q}

for

r

\geq 0,

_p

\geq 0,

_q \geq 1

with

_{(1+2r)q\geq(p+2r)}

,Proc. Amer. Math.

Soc.,

101

(1987),

85‐88.

[6]

T.

_Fmuta,

Generalized

_{Aluthge transformation}

on_p

‐hyponormal

_operators,

Proc. Amer. Math.

_Soc.,

124

_(1996),

3071‐3075.

[7]

E.

Heinz, Beiträge

zur

Störungstheorie

der

Spektralzerlegung,

Math.

Ann.,

123

_(1951),

415‐438.

[8]

M.

_Ito,

Some classes

_of

_operators associatedwith

_{generalized Aluthge}

trans‐

[9]

M. Ito and T.

_Yamazaki,

Relationsbetween two

_inequalities

(B^{r}\mathrm{z}A^{p}B^{\mathrm{r}}\mathrm{z})^{\frac{f}{p+r}}

\geq

B^{r} and A^{\mathrm{p}} \geq

(A^{\frac{p}{2}}B^{r}A^{\mathrm{p}} $\Gamma$)

and their

_{applications, Integr. Equ. Oper. Theory,}

44

_(2002),

442‐450.

[10]

I.H. Jeon and B.P

_Duggal,

On _operators with absolute value

_conditions,

J.

Korean. Math. Soc

_{41(2004),617-627.}

[11]

Jan

_{Stochel., Seminormality}

of _operators from their tensor

_products,

Proc.

Amer. Math. Soc

_{124(1996)435-440.}

[12]

K.

_Löwner,

Über

monotone

_{Matnxhnktionen,}

Math.

_Z.,

38

_(1934),

177‐216.

[13]

K. Tanahashi and M.

_Chō,

Tensor Productsof

_{{\rm Log}}

_{‐hyponormal}

and of Class

A(s,

t)

operators,

Glasgow

MathJ.

_46(2004)

91‐91.

[14]

A.

_Uchiyama,

K. Tanahashi and J. I.

_Lee,

_{Spectrum of}

class

_A(s,

_t)

_operators,

Acta Sci. Math.

_(Szeged),

70

_(2004),

279‐287.

[15]

M.

_Yanagida,

Powers

_of

class

_{wA(s, t)}

_operatorswith

_{generalised Aluthge}

trans‐

formation,

J.

_Inequal.

_Appl.,

7

_(2002),

143‐168.

[16]

T. Prasadand K.

_Tanahashi,

On class

_p‐wA

_{(s, t)}

_operators, FunctionalAnal‐

ysis, Approximation Computation,

6

_{(2) (2014),}

39‐42.

[17]

T.

_Yshino,

The_p

_{‐hyponormality of}

the

_{Aluthge transform, Interdiscip.}

Inform.

Sci.,

3

_(1997),

91‐93.

[18]

J. YuanandC.

_{Yang, Spectrum of}

class

_wF(p,

r,

q)

operators

for

p+r\leq 1 and

q\geq 1, Acta Sci. Math.

(Szeged),

71

(2005),

767‐779.

[19]

C.

_Yang

and J.

_Yuan,

On class

_wF(p,

r,

q)

operators, Acta. Math.

Sci.,

27

(2007),

769‐780.

M. Chō

Departament

of

_{Mathematics, Kanagawa University,}

Yokohama

_221‐8686,

Japan

E‐mailaddress:

_{chiyom01@kanagawa‐u.ac.jp}

T. Prasad

Schoolof

_Mathematics,

Indian Institute ofScienceEducationand Research‐

Thiruvananthapuram, Thiruvananthapuram‐695016, Kerala,

India

E‐mail address:

_{prasadvalapil@gmail.com}

M.H.M.Rashid

Department

of

_{Mathematics, Faculty}

ofScience

_P.O.Box(7),

Mu’tah univer‐

sity,

Al‐Karak,

Jordan

E‐mail address:

_{malik‐okashaOyahoo.}

com

K. Tanahashi

TohokuMedical and Pharmaceutical

_University,

Sendai

_{981‐8558, Japan}

E‐mail address:

_{tanahasi@tohoku‐mpu.ac.jp}

A.

_Uchiyama

Department

ofMathematical

_{Science, Faculty}

of

_{Science, Yamagata}

Univer‐

sity

Yamagata, 990‐8560, Japan