AND WORDS

VOL. 21 NO. 3 (1998) 565-570

SOME PROPERTIES OF

PREREFLEXlVE

SUBSPACES

OF

OPERATORS

JIANKUI LI

Department

ofMathematics

University ofScience andTechnologyof China Hefei,Anhui230026,P.R. China

(Received May 31, 1995 and in revised form December i0, 1995)

ABSTRACT.

In

the paper, ^we define a notion of prereflexivityfor subspaces, give several equivalent conditionsofthis notion andprove that if^$ C_

L(H)

^isprereflexive, then every a- weakly closed subspace ofSis prereflexiveifand onlyif$ hastheproperty

WP(see

definition

2.11). By

ourresult, weconstruct areflexive operatorAsuch that A 0isnot prereflexive.

KEY WORDSAND PHRASES: Prereflexivesubspace,reflexiveoperator.

1991 AMS SUBJECT CLASSIFICATION CODES: 47D15,47D20.

I. INTRODUCTION

The concept ofreflexivity foralgebrasof operators was introduced by Halmos

[1].

^There

is anatural generalizationwhichwasfirstformulated by Loginov and Sul’man

[2].

^Arveson

[3]

introducedthe concept ofprereflexivityfor

algebras

^butnothing correspondingtothishas been studied in the generalized version. The concept of prereflexivity has already proveditsworth.

In this paper, we define a notion ofprerefiexivity for subspaces ofoperators which extends theconcept ofprereflexivityforalgebras.

In

Section 2, wegive several equivalent conditionsof prereflexivity forsubspaces,prove that ifSis aa-weaklyclosedsubspace,then^,5’has theproperty WPif andonly ifSishereditarily prereflexivein thesensethat everya-weaklyclosedsubspace ofSis prereflexive. InSection3, usingtheresults in Section2, weconstructaprereflexivebut not reflexive operator andprove that thereexistsareflexiveoperator

A

such that

A

0is not prereflexive.

Throughoutthe paper, let

H

denoteacomplexseparableHilbert spaceand let

L(H)

^denote

the algebra ofall bounded linearoperatorson H. Wewrite

T(H)

for the ideal of trace class operatorsin

L(H), F

for the finite rankoperatorsin

T(H)

^and

Fk

for the subset of

F

consisting of operators of rank kor less. The trace norm is denoted by

[[. [].

IfS ^C

L(H),

^{we denote} S+/-forits preannihilator, i.e.,

S.L {t

E

T(H) tr(at)

^{0for allaE}

S};

^{dually,thenotation} .M^+/- indicates the annihilator ofa subset

.M

of

T(H),

^{that is} .M^"L

{a

^E

L(H) tr(at)

0for all aE

.M}. ^For

any

A

E

L(H),

the symbol A^(’0 denotesA

...

^{@A. IfS isasubset of}

L(H),

$(") denotes

{A

⁽ⁿ⁾

A

E

S}.

Forany x,y in

H,

let x(R)y denote the rank-1 operator u^---,

(u,z)y.

^Let beacollectionof

(closed linear)

subspaces of

H,

algdenotesthe set ofall operatorsacting^onHthatleave every member of invariant. Dually,if isasetofoperators acting ^on

H, lat

^denotes the collection ofsub,paces of

H

^which are left invariant by every member of

2. SOME RESULTS OF

PREREFLEXIVE

SUBSPACES

In

[3,4],

Arvesonintroducedthefollowingconcept ofprereflexivity foralgebras.

DEFINITION2.1. Aa-weakly closedalgebra

A

C_

L(H)

^iscalled prereflexiveif

ANA"

alglaA D(alglatA)*.

DEFINITION2.2. Aa-weakly closed subspace of

L(H)

^iscalled n-prereflexiveifwhenever

T L(H (’’))

^satisfies the condition that

Tx [,5’(’)x]

^and T*x

[S’0x]

^forallzin

H

then T

isin

S(n).(Here [.

denotesnormclosed linear span.) When referenceto nisomitted,itisunderstoodtobe1.

REMARKS. Since

L(H)

^isn-prereflexive, toprqve that Sis n-prereflexive we needonly to prove that whenever

T

6

L(H)

satisfiesT(")z ₆

[S(")z]

^and

^T(’)*z

⁶

[S(")z]

^{for allxin}

H

^(") then

T

is inS.

By

the definition2.2,weeasilyprovethatif

U

isaunitary operatorin

L(H)

^{then USU* is}

prereflexiveifand onlyifS isprereflexive.

PROPOSITION 2.3. Aunital a-weakly closedalgebra^,4is prereflexiveas a subspaceif andonly ifit isprereflexiveasanalgebra

(i.e. A

Cl

A*

(alglatA.)*Clalglat.A).

PROOF. Supposethat

A

isprereflexiveas asubspaceofoperators. LetT (alglatA)*^Cl alglatA. Then we have that for any M latA,

TM

C_ M,T*M C_ M. For any z

H, [Az]

elatAand

I e A,

wehavethat T*x

e [Ax]

^and

Tz e [Ax]. ^By

prereflexivityof

A

^{as a} subspace, wehave that

T A

andT*

A,

thus

A

^ClA* D_ (alglatA)* atglatA. The^reverse inclusionalwaysholds,hence

A

isprereflexive as analgebra.

Conversely,letTx

e [Az]

^andT*z

[Ax]

^{for allz}

e

^{H. Then}

TM

C_

M,

T*M C_

M,

VM latA. Since

A

isprereflexiveas analgebra, wehavethat

T A.

Hence

A

is prereflexive as a

subspace.

Q.E.D.

If is anarbitrary subset of

L(H),

^{thenwe use}preref(cp)todenote the closure of

span{S,

T"

S

,T L(H),Tz e [ox]

^and T*x

e [z]

for all x

H}

^{in a-weakoperator} topology. It follows thatpreref() is thesmallest prereflexive subspace containing

,

^{and is} prereflexive i]and onlyif preref().

PROPOSITION 2.4. IfSisaa-weakly closed subspace of

L(H),

^{thenS is} prereflexive ifand onlyif

prefer(S)

^gl

(prefer(S))* tel(S)

¹

(tel(S))*

S1S*.

PROOF. The necessity istrivial,so wehaveonlyto prove the sufficiency.

If

T L(H),Tz [Sx]

^andT*x

[Sz],

^so

T prefer(S)

Cl

(prefer(S))*

SNS* C_ S.

Hence Sisprereflexive.

Q.E.D.

By

the previous proposition, weget that ^{,5" is}prereflexiveifand only if,5"* is prereflexive;

and ifS isaunitalalgebra, Proposition2.4is theanalogy of thedefinitionofprereflexivity for unitalalgebras thatArvesongives.

THEOREM 2.5. IfS is aa-weaklyclosed subspace of

L(H),

^{then Sis} n-prereflexive if andonly if

S+/-C_

span{(S+/-

^U

S+/-*)

^gi

F,}

^I1"11’

PROOF. Ifrank

f ^<_

^n,wehavez,,...,x,,y,,...,y,inH such that

f

x(R)y,+...+z,(R)y,.

LetT

L(H),

^then

tr(Tf)= (Tyi, xi)= (T("),5)

where x, x,,,

"

^{y, (R) (R)}

i=l

y,,5

and

"

ⁱⁿ

^H

("). Hence

f

^{S+/- ifandonlyif}

(S(")’, ")

^{0 for all S S if and only if}

"

⁶

[S(")y-’]

+/-. So

tr(Tf)

O,

tr(T*f)

⁰

tr(Tf*)

^{forall fin}

S+/-

^withrank

f ^_<

^{nifand only}

if

T(") [S(")y-’]

and

T(")*" [S(")y,

^forall

"

^inH(").

IfSisn-prereflexive, theaboveparagraphshows

span{(S+/-

U

S+/-*)

^Cl

^F,, }+/-

^C_

Hence

S+/- ^C

span{(S+/-

U

$+/-*)

^I"1

F,.,}

’11"11.

Conversely,ifS+/- C_span{(S+/-U

S+/-*)

^Cl

F,}

^{I1"11’ letT}

L(H)

such that for any

H("),T(") [S(n)y-’], T(n)* [S().

^Then

t,’(Tf)

O,

tr(Tf*)

^0,for any

f

^{,5’+/- withrank}

f ^_<

^{n, so}

T (span{(S+/-

U

S+/-*)Cl F.} II,)+/-

C_S.

Hence Sis prereflexive.

Q.E.D.

By Theorem 2.5, we have that ifS is self-adjoint, then S isreflexive if and only if^,9 is prereflexive.

COROLLARY 2.6. IfasubspaceSof

L(H)

^isn-prereflexive,thenit ism-prereflexivefor re>no

PROPOSITION 2.7. For i,j 1,...,n, let S, bea a-weakly closedsubspace of

L(H)

andlet Sbe thesubspaceof

L(H("))

^definedby

s {(,,),,x. ,,

^E

ThenSisprereflexiveif andonlyif

span{(S,,+/-

^U

S,.L"

^F

F }ll-II,

_D

Su.

L.

PROOF. For$+/-

{(au),x, a,

6

Sj,+/-},

by Theorem 2.5,wehave that,5" isprereflexive ifandonlyif

span{(,gu+/-

^U

8,+/-’)

^F

F} IIII

_{_D}

u+/-" ^Q.E.D.

COROLLARY 2.8. Let

Si2 ^{(l _< _<}

^j

^{_< r)}

^{bea-weaklyclosedsubspaceof}

^L(H),

^define

all a12 aln

0 _{a22 a2n}

S=

la,jESu,l<i<j<n

0 0 a,,

ThenSisprereflexiveifandonlyif everyS,, isprereflexive.

PROPOSITION 2.9. Let S $1(B @

S,,

whereS, is a a-weaklyclosed subspace^of

L(H,).

^ThenSis aprereflexive subspaceof

L(HI @...(BH,)

ifandonlyif everyS,isprereflexive.

Theproofiseasy, weleave theprooftothe reader.

PROPOSITION 2.10. Let Sbe aa-weakly closed subspace of

L(H),

define below the subalgebraof

L(H

^E)

H)

A=

iI

[AEC,sES

Then

A

isprereflexiveifand onlyifS+/-3

F #

^0.

PROOF. Supposethat.,4isprereflexive. IfS+/-^f3

F1

0, wehavethat forallz 6

H,

z

O, [Sz]

^{H. For}y

()

⁶

^{H (2),}

^{if y 0,letb,} ₀

_I

^{if y 0, te}

^an

^{ESsuch that}

,-lima"y=x’letb"= (O0 a,)o

Ineitherce, wehavethat

,-limb"= ( IO 00)"

^SinceA

isprereflexive,wehve that 0 0

Conversely,byCorolly 2.8, wehave he

A= I

^{A,6C, sES}

(, 0)

is prereflexive, sopreref(A) C_

f.

In thefollowing, we prove that

0 0 preref(A). By S+/-FF,

#

^{0,weget that there existxandy in}

^H

satisfying^that

[[x[[ [[y[[

1,z (R)y 6 S+/-,

( ) ( )

hence

r/=(_)(R)(;)e.A+/-.

^Since

^{tr(r/ I}

₀ ⁰₀

#

^0,wehavethat

^I

₀ ⁰₀

q

preref(A). Hence

A

isprereflexive.

Q.E.D.

In

[5],

weprove that ifSisaa-weaklyclosedsubspaceof

L(H),

andwelet

A: ^AI

^{0 00 s0}

,4=

.. ^I,\EC,

^s6S

0 0 )I 0

0 0 0

AI

wheren

>_

3, then

A

isprereflexive.

By Propositions 2.7, 2.10 and Proposition 3.10

[6],

^we know that the reflexivity is very different to theprereflexivity. Let S beaprereflexive subspaceof

L(H).

^{ThenSis saidto be} heredztarzlyprereflexzveif everya-weakly closed subspace ofSisprereflexive. Inthe following wediscusshereditary prerefiexivity.

DEFINITION 2.11. Let S beaa-weakly closed subspace of

L(H).

^{We say that S has}

thepropertyWP ifitstatistics

(S_L + F1

^U

(S+/- + span{(S.L

U

S+/-*)

^gl

FI }i1"11,) T(H).

REMARK. Theproperty WPisaproperty^whichisweakerthan theproperty P.

THEOREM 2.12. Let S be a prereflexive subspace of

L(H).

^{Then S is} hereditarily prereflexive if and only ifS hasthe propertyWP.

PROOF.

Suppose

thatS hasthe property WP. Let 12 beanya-weakly closed subspace ofS. Forany inY+/-C_

T(H),

^{weconsiderbelow thetwo cases:}

(i)

^If 6S+/-

+

F1, then t=f

+

^gwith

f

^{6 S+/- andg 6F1,g}

f

⁶

12.L I"1F.

^{Since Sis} prereflexive,wehave

f

^{6 S+/- C_}

^span{(S+/-

^O

^S+/-*)

^N

F1 }11"111

C_

span{(]]+/-

U

V+/-*)

^N

F1 }11"11,.

Hence

v,{(v. u

v_’)

n

F,

(ii)

^If

e S + span{(S+/-US+/-*)nF,} III1’,

for S+/- C_

span{(S+/-US+/-’)nF1}

^{III1’ C_}

,r{(v. u v+/-’) n

F,

}11.1,

^wehave

e

^span

(12+/-

^U12+/-*

n Fa ^,

Bythe abovetwo cases, wehave that 12+/- ^C_

span{(Y+/-

^U

12+/-*)n Fa }ll.ll,.

^By Theorem 2.5, wehavethat 12isprereflexive.

Conversely, suppose that

(S+/- + F)

^U

(S+/- + ^span{(S.L

^U

^S+/-’) 91Fx} I1"11’) _{# T(H).}

Let

q (S+/- + ^F)

^U

^(S+/- + span{(S+/- US+/-*) I"1F }11.11,)

^but

e T(H)

^{and define}12

(Ct +S+/-) +/-,

wehave that 12 is aa-weakly closed subspace ofS. In thefollowing we prove that I) isnot prereflexive. Since1)+/-I"1

F1 S+/-

⁹¹F1,wehave

(V. u _V+/-’)

n/5

(S+/- u S+/-’) n F.

Suppose 1) isprereflexive. Wehave

V_

{(S+/- uSz’) n F}

^+/-

{(VuV.’)nF} +/-,

then12+/- Ct

+ ^S+/-

^C_

^span{(S+/-

^U

S+/-’) n FI }11-11,.

^{Itisimpossiblesince}

^(S+/- + ^F)

^U

^(S+/- +

span{(S+/- US+/-’) n F1}II’IIt). Q.E.D.

PROPOSITION 2.13. Let Sbeaweakly closedsubspaceof

L(H)

^suchthat

(S+/- + Fk)

U

span{(S+/-

U

8+/-’) n r2&+l}

^1]’11’

T(H).

ThenSis

(2k+

1)-prereflexive.

PROOF. SinceS isweakly closed,itfollowsthatS+/-

n F

^I1"11 S+/-. ByTheorem 2.5 we

onlyneed to provethat

span{(S+/-

U

S+/-*) n Fk+a

^{D_ S+/-. Since S+/-fl}

^F U ^(S+/-nF,),

^it

sces

to provefor all

>

2k

+

^1, il

s+/-

n g span{(,9+/-

^U

S.L’) n F+ }.,.11,.

(9_.

1)

Ifwe canshow

span{(S+/-

U

,.q+/-*) n

F,_, ^II.n,

span{(S+/- u S+/-’)

^rl

Ft}

^I1"11’

with

>

2k

+

^{1, wehave that}

(2.1)

^{is true. Let E}

(S.L

^O

S.L*)F

^{F, with}

>

^2k

+

^1,wemay

assume that E

S.L

^F

F(if S.L

^gl

F

^wemay consider

t*),

^write t=f+g with

f

^E

F+I

^and

g

F-k-1.

^Byhypothesis,wehave

f,g

e (S_L + Fi)

^Uspan{(S.L^L3

S.L*) F2:+i

If

f

^E

S.L + F,

wehave that there exists an hin

Fi

^{such that}

/

h E

S.L, f

^h

+

g

+

^h.

Since

f

^{h E}

S.L

FF2+1, g

+

h EF_IF

S.L,

itfollows that E

span{(S.L

U

S.L’)

^F

F,_I I[.11,.

Similarly,if gES+/-

+ F,

^wemay prove that

E

span{(S.L

U

S.L’)

^AF_I

)]1.]1,.

If

ft S.L +

^{F, and g}

S.L +

^{Fk, we have that /,g E}

^span{(S.L

^U

S.L*) nF2k+x} ll’II’.

Hence f+g E

span{(S

U

Sx*) F2t+}

^ll’ll’ span{(SU

8*) Ft_l} II’ll’. Q.E.D.

PROPOSITION 2.14. Let S beawetly closed subspaceof

L(H)

satising that given z,...,x,

H,

thereexistsx E

H

such that

IITx, llTxII,

^forM1

T

^{$. Then every}wetly closed subspaceofSisprereflexive.

Theproof^isey,weomit it.

3. AN APPLICATION.

IfA

L(H),

^let

w(A)

denote the closurein theweak operatortopology^of

L(H)

^{ofthe set}

ofpolynomiMsin Aand

I,

let

wo(A)

denote thewetly closedprincipal ided generated by A.

AnoperatorAis calledprereflexiveif

w(A)

^isprereflexive. In

[7],

^Larson=d

Wogen

construct areflexive operator A

su

that A@ 0 isnot reflexive.

In

thesection, anpplicationof the resets in Section 2, ^we prove that there exists a reflexiveoperator A

su

that A@0is not prereflexive. Bythe idea in

[8],

^westconstructaprereflexive butnot reflexiveoperator.

Let

H

beaseparableHilbertspace ofmeionj d let

K @ ^H.

Consider theHilbert spe

K

@H. If1 k

< ,

^let

P

^{be theorthogonal}projection ofK

H

ontothe k

*

suand of

H

inKd let

P

^{be theprojectionof}

^K

^@

^H

^{onto0@ H. Fory}

^T

^E

^L(K

^@

^H),T

^admits

atrix representation

T (Ti1)IS,,Sm,

^with

Ti1

^E

^L(H).

If

A L(K@H)

^let

A, P,P,

^wemay choose toview

A,

^{either asubset of}

or asubset of

L(H).

^Fory

L(H)

^let

[]u {S

E

L(KH) S,,

^E

d

^S,

0if(k,l) #

(i,j)}. Let

= ^{A-Ax,

^{A E}

^A}.

^{Let @ bed}wetly closed sub,pace of

L(H)

^{such that}

(2)

^isprereflexive but not reflexive.

By

Proposition3

[8],

^{wemayconstruct}

operator

T su

that

w(T <)) w(T<) 4

By Lemma6

[8],

^wehave

w(T(2)

^isreflexive. Since

v

^{() isnotreflexive,itfollows}

^w(T (2))

^isnot

reflexive.

In

thefollowingweprovethat

w(T ())

^isprereflexive. Sincepreref(w(T

[preref((2))],oo.

wehave that ifA E

L(K

^{(2) H}

(2))

^{such thatfor}

[w(T(2))xl,

A’z E

[w(T(2))z],

^then A

A 4

^{A, where}

Ax ^e w(T(2),

^d

A2

_{0 0} satisfyingthatforyy E

H (), Axy

E

[(2)y],Ay

E

[9()y].

Since

(2)

isprereflexive, we have

Ax

^E

(2),

soAE

w(T(2) 4 [(2)]1,. ^Hence w(T (2))

^is prereflexive.

PROPOSITION 3.1.

Suppose

that

H

^d e Hilbert spaces with

dim

1. Let A

L(H)

^{andlet 0}E

L().

(1)

A

0

^isprereflexive if andonly if

wo(A)

^isprereflexive.

(2)

^{IfA is} prereflexive, then A 0 is not prereflexive ifd only if

I wo(A)

^but

^I

^E preref(wo(A)).

PROOF.

(1)

^Let

BEL(H@[-I) suchthatforanyx@yeH$

B(x

^y)E

[w(A

3

0)(z

3

y)], (3.1)

B’(x

y)E

[w(A 0)(z y)], (3.2)

wehavethatBE

prere/((Ae0)).

For preref(w(A))@CI^isprereflexivedcontMns

(Ae0),

itfollows that B

B ^AI,

^where

B

^E preref(w(A)). It suffices to provethat

B1 ^AI

^E

wo(A),

^since

w(A O) wo(A 0) + C(I I) wo(A)

⁰

+ C(I I).

Forafixed nonzero vector y in

K

d for yxin

H,

by

(3.1),

^wehaveasequence ofpolynoMs

{p,}

such that

2ip=(A ^O)(x

^y)

^{(B1 AI)(x}

^@

^{y) Bxx Ay.}

^Sincep=(A

^O)

^{p=(A)@p,(O)I, thus}

Let q p,-p(0), then

%(0)

O, qn(A)x (B AI)x, that is

(B AI)x [0(A)].

By

(3.2),

^we may prove that

(B; I)x [wo(A)x].

^Since

w0(A)

^is prereflexive, we have

B ^AI 0(A).

Conversely,suppose that

w(A. 0)

isprereflexive. Let T preref(wo(A)). For

=(A, 0) 0(A, 0)+ C(Z Z) 0(A)

0

+ ^{C(Z, Z),}

it follows thatT

wo(A).

Hence

wo(A)

^isprereflexive.

(2)

^Supposethat Aisprereflexive. For

o(A) **Y(0(A)) g ,,((A)) (A) 0(A) +

^CL

(3.3)

By

(1),

wehave that A 0isnotprereflexiveif=donlyif

wo(A)

^isnot prereflexive.

By (3.3),

itfollows that

w0(A)

isnotprereflexiveifandonlyif

I

$

wo(A)

^but

I preref(wo(A)). Q.E.D.

By

Proposition 2.1 =d Theorem 3.7

[7]

^{well the the}above proposition we have the followingresets.

COROLLARY3.2. IfAisreflexive,thenA0^isreflexive if donlyifA@0^isprereflexive.

COROLLARY 3.3. ThereexistsareflexiveoperatorA

su

thatA 0isnotprereflexive.

REFERENCES

1.

HALMOS, P.R.,

Reflexive latticesof subspaces,J.London Math.Soc.

(2)

⁴

(1971),

^257-263.

2.

LOGINOV,

A.Iand

SUL’MAN, V.S.,

Hereditaryandintermediatereflexivity ofW*-algebras, Math. USSR- Izv 9

(1975),

1189-1201.

3.

ARVESON, W.B., Operator

algebras and invariant subspaces, Arm.

of

^Math.

(2)

¹⁰⁰

(1974),

433-532.

4.

ARVESON, W.B.,

Tenlecturesonoperatoralgebras, CBMSseries 55, Amer. Math. Soc., Providence

(1983).

5.

LI JIANKUI,

Someproperties ofprereflexivealgebras, J. QufuNormalUniv.

(2)

¹⁶

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