^o

REFLEXIVE LATTICES OF SUBSPACSS IN A LOCALLY CONVEX SPAC£

by

ALICE (HIETCHEN MILLER MOONINGHAM, B*A*, M.A*

A DISSERTATION

IN

MATHEMATICS

Submitted to the (k>aduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Accepted

August, I97U

AB^-55 2 1

f \^

SO

Cop 7.

ACKNOWLEDGMENTS

I am deeply indebted to Professor T. G. Nevman for

his guidance and instruction in the preparation of this

dissertation and to the other members of my committee.

Professors G. L. Baldwin, H, R. Bennett, C. N. Kellogg,

and J, D. Tarwater, for their helpful criticism*

ii

CONTENTS

ACKNOWLEDGMENTS ii

INTRODUCTION

CHAPTER I. General Properties of Reflexive Lattices ... U

CHAPTER II. Finite Dimensional Results 13

CHAPTER III. Weakly-Reflexive: Mackey-Reflexive 36

CHAPTER IV. Complete Chains kk

CHAPTER V. A Class of Reflexive Lattices U6

CONCLUSION 53

LIST OF REFERENCES 55

iii

nrrooDucnoN

In the context of a tox>ological vector space the

invariant subspace problem asks vhether the set of all

invariant subspaces of a continuous linear transformation

can consist of the two extremes only, the space itself

and {0} • An even more interesting problem is encountered

in attempting to describe the set of invariant subspaces

of a single transformation, or a set of transformations*

Similarly, given a set of subspaces, vhat continuous linear

transformations leave invariant each of the subspaces in

the set?

Much vork has been done in the area of invariant sub-

space lattices, or reflexive lattices. In [5l and [6],

P. R. Halmos summarizes many of the findings. Complete

chains of closed subspaces in Hilbert space are reflexive,

as vas shovn by J. R. Ringrose [11]* K* J* Harrison [7]

has described a rather large class of reflexive lattices

of closed subspaces in Hilbert space, which includes finite

distributive lattices and complete atonic Boolean algebras

of subspaces.

All of this vork has been done in the context of

Hilbert space* ThiB paper attempts to examine the problem

from the more general point of viev of certain locally

convex, topological vector speuses.

In Chapter I, reflexive lattices are defined and some

generaa properties are investigated. It is seen that the

collection of all reflexive sublattices of closed subspaces

itself forms a lattice.

In the second chapter our attention is restricted to

the finite dimensional case. When ^ C is reflexive then

A l g ^ is semisimple vith minimum condition if and only if

j i is atomic. The idea of a maximal reflexive lattice is

introduced, euid it is seen that every reflexive lattice is

contained in a maximal reflexive lattice. Also, any re­

flexive lattice may be realized as a meet of meet-irre­

ducible reflexive lattices. This becomes apparent as

soon as it is realized that the lattice of reflexive

lattices satisfies both chain conditions. Finally, it

is seen that in order to describe the invariant subspace

lattice of an eurbitrary linear transformation, it is

sufficient to examine the invariant subspace lattices of

nilpotent linear transformations and of semisimple linear

transformations. The remainder of the chapter is devoted

to this task.

In Chapter III it is shovn that the veakly-reflexive

sublattices of closed subspaces and the Mackey-reflexive

sublattices of closed subspaces coincide* Thus, it is

seen that in Banach spaces, vhere the norm topology

and the Mackey topology coincide, questions of re-

flexivity may be resolved by using the veak topology

only.

The techniques of duality, as described in

Chapter III, prove very useful in treating questions

of reflexivity in locally convex topological vector

spaces for vhich there is a duality. In Chapter IV

these techniques are utilized to show that coisplete

chains of closed subspaces are reflexive; vhile in

Chapter V complete lattices of closed subspaces, con­

taining the trivial subspaces, vith the property that

the lattice is infinitely meet-distributive and the

property that every non-zero element of the lattice is

a Join of completely Join-irreducibles are shovn to be

reflexive.

Throughout the text, lemmas, definitions, propo­

sitions, and theorems are labelled consecutively by

chapter. For example. Lemma 2.13 is the thirteenth

labelled item in Chapter II. Well-knovn definitions

and theorems are not set apart; but rather are included

in the text vithout any special labelling* In order to

simplify notation A C B vill mean A is a proper subset

of B; A C B vill mean A is a subset of B, possibly

equal B.

CHAPTER I

GENERAL PROPERTIES OF REFLEXIVE LATTICES

Let E and E* be complex vector spaces. To say that

E and £' form a dual pair [^], denoted (E,E*)9 means that

there exists a non-degenerate bilinear form, <,> , mapping

E X £• to f, the complex numbers. If such a form exists,

E and E' are said to be dual vith respect to <,> •

The weak topology [12] on E, denoted by a(E,E*)» is

the coarsest topology on £ such that for each x* e E' the

mapping x *»> <x,x'> is continuous. For each x' e E', let

S , « {x I X e E, <x,x'> £ 1} . It turns out that

{S . I X* e E* > is a subbase for the closed sets at zero. x'

With the weak topology, E is both locally convex and

Hausdorff.

If M is a subset of E, the polar of M, denoted M^,

consists of (x* e £• | <x,x'> < 1, x € M} . When M is a

subspace, then M, the topological closure of M; M ; and

M , the orthogonal complement of M are also subspaces*

In addition, when M is a subspace, M^ « M^ and

M « M*^ » M°^ [12].

The set of all linear functioncds on E is called the n

algebraic dual of £ and is denoted by £ [12]* When (E,E*) h

is a dual pair, consideration of the linear functionals

f , on E with f^,(x) » <x,x*> shows that E* is isonorphic x* ^

to a vector subspace of S . Hiat is, E» is isomorphic to

{f i |x* c £• >, and each f i is a(E,E« )-contiimou8*

Bius the dual of E under a(E,E«) is £• itself.

If (£,£• ) is a dual pair and x is a locally convex

topology which is coopatible with the linear structure,

T is said to be compatible with the duality provided each

lineeu: functional, x •*' <x,x'> vith x* e E', is continuous;

and conversely, ea^h continuous linear functional is of

this form. Mote that o(E,E*) is the coarsest topology

which is cosq;>atible with the duality. The Mackey topology,

denoted m(E,£') is the finest topology which is coo­

patible with the duality. One important property connon

to these topologies is that the closed subspaces are the

same in all topologies compatible with the duality [12].

In all that follows, unless otherwise specified, all top­

ologies will be assumed to be compatible with the duality.

For each x e E and x* e E', define the linear mapping

xQx* from E to Eby x ® x ' (z) « <z,x'> x for each z e E.

Note that x 9 x* is continuous in any topology compatible

vith the duedity* One characteristic of these maps, which

will be utilized extensively, is the following: if M is

a subspace of E and x ® x* (M) C M, then either x e M or

X* e M *

Let P and Q be peseta vith ^ : P •»> Q and f : Q^ P

correspondences such that

1*) X <.x» implies •(x) >.*(xM for x,x» e P,

11.) y ly* inplies ^Cy) >, •(y») for y,y» c Q,

iii.) X < ••(x) and y £ ••(y) for xe P, y c Q*

The correspondences i and ^ are said to define a Galois

connection betveen P and Q [l]* The concept of Galois

connection vill play an important role in the description

of the operations of "Alg" and "Lat" vhich vill nov be

introduced*

Let E denote a loccdly convex topological vector

space* For each set J^ of closed subspaces of E, A l g ^

vill denote the set of all continuous linear transfor­

mations on £ vhich leave invarinat each subspCLce in ^ *

Also, for each set ^ of continuous linear transformations

on E, Lat£ vill denote the set of all closed sub-

spaces of £ vhich are invariant under each transformation

in ^ * It is easily seen that for any set iC of closed

subspcuses, A l g ^ is an algebra; and for any set K of

continuous linear transformations, Lat^ is a lattice*

For information concerning these operations see [5] and

[6]* Furthermore, it may be verified that **Alg** defines

a correspondence from the poset of sets of closed sub-

8i>ace8 of E to the poset of sets of continuous linear

transformations on E* Similarly, "Lat** exhibits a

correspondence betveen the poset consisting of sets

of continuous linear transformations on E to the poset

consisting of sets of closed subspaces* Deeper exam­

ination reveals that "Lat" and "Alg" are order reversing

operations and

^ C Lat Alg;«f and ^ C Alg Lat *

Thus "Lat" and "Alg" are correspondences vhich define

a Gcaois connection between the poset consisting of sets

of continuous linear transformations on E and the poset

consisting of sets of closed subspaces of £*

A lattice iC with the property that Lat Alg i/L ^ jt

is of particular interest*

Definition 1.1: A lattice ^ of closed subspaces of

E is a reflexive lattice if X * Lat Algjc . An algebra

^ of continuous linear transformations is a reflexive

algebra if 4 » Alg L a t ^ .

Note that each reflexive lattice contains the trivial

subspaces (O) and E; and the lattice consisting of all

closed subspaces of E is reflexive. Also, ^ L a t Alg;o

implies A l g ^ ^ Alg Lat Alg ji . On the other hand,

Alg;tf o Alg Lat Alg;^ , by property iii*) for Galois

connections. Therefore, A l g ^ « Alg Lat Alg >u *

Similarly, it may be shovn that L a t ^ « Lat Alg L a t ^ •

Oms, Alg^ and Lat^ are reflexive algebras and lattices,

respectively*

When confusion may arise as to the topology in

question, the following notation vill be employed:

A l g ^ vill denote the set of all T-continuous linear

transformations on E vhich leave invariant each subspace

in*C . Similarly, Lat ^ vill denote the set of all

T-closed subspaces of £ which are inveuriant under each

operator in ^ . When the topology in question is o(E,EMt

then Alg ^ will be shortened to Alg*f* Also, if the

a(E,E«) ^ topolo^ is a(E',E), then Alg JC will be designated

o(E»,E) by Alg^. Similarly, for the Mackey topology, we will

a» use the abbreviated notation Alg and Alg * Note that

m m'

for topologies compatible with the duality, since the

closed subspaces coincide, there is no confusion in

using Lat with no specification of topology.

The remfldnder of this chapter will be devoted to

examining some general properties of reflexive lattices.

Proposition 1*2: A reflexive lattice of closed subspaces

is complete*

Proof: Suppose gC is reflexive and L e X • a tjl» — — o '

First it must be shown that ^ L c X * Ifxe ^ L

and A e Alg^ , then A(x) e L for each a ej^ , vhich a

implies that X. is meet-complete*

8

Next to see that ^V^ L tX » let A c Alg/,

a n d x e V ^ L • < L > . There exists a net

(x } such that {x^} converges to x vith x^ e L for Y Y "Y

some o e ^ * Since A is continuous,{A(x )} converges ' Y

to A(x)* But because A e Alg at implies A(x ) c L ,

then A(x) z y "L^ * a zX

Y <* ' Y

Proposition 1.3: Lat Alg^ is the smallest reflexive

lattice containing tH .

Proof: Suppose X is a lattice of closed subspaces

and ^ » is a reflexive lattice such that C Jf'cLat Alg,|f*

Since Alg/ B ^Ig ;t' Alg Lat Alg^ = Algjl^ , then

A l g ^ « Alg d,\ Thus Lat Alg^ = Lat Alg id • « ;6'

since ^ is reflexive.

The set of CLLI reflexive lattices itself forms a

lattice. Given tvo reflexive lattices , and ^-,

their Join, ^ \j V , vill be defined to be the

smallest reflexive lattice containing tk^ and «^p. The

meet operation may be taken to be the intersection of

/ and ^ , €is the next proposition verifies.

Proposition 1*U: If ^ and JL are reflexive, then

«C ^^^ •Cp is reflexive.

Proof: It must be shovn that

/.^n ;^2 ' ^ * A l g ( ^ ^ / ) X^^). Since alvays

^ l O j i f a - ^ * ^^^ *^i/O ^ 2 ^ » * ^^ ^^"

ficient to shov that Lat Alg( / ^ /) /. 2^ S /,i O jfa'

For i « 1,2, since ^^"^/^ /I c f , then

j^^ = Lat Alg _^5 Lat Alg( / ^ f) Ji^). Thus

Pt'oposition 1.5: Let {U } be a family of closed a o c^

subspaces of E. Suppose <U > is the sublattice generated

by {U > . Then Lat Alg {U } » Lat Alg <U > . a a a

Proof: Since (U^) ^ <U > , then

Lat Alg {U } <i Lat Alg <U > . o — a

On the other heuid, since {U } Lat Alg {U } , then o ^ ex

< U > 5 Lat Alg {U } * a a

Thus < U > C Lat Alg {U > Cl Lat Alg < U > so that o "" a "" o

Lat Alg <U > C Lat Alg Lat Alg {U } « Lat Alg{ U > o "• a *

C Lat Alg Lat Alg< U > « Lat Alg< U > . — o a

Consequently, Lat Alg {U } = Lat Alg < U > . o o

For each lattice ^ of closed subspaces of E vith

(£,£*) a ducd pair, let ^* be defined as follows: 1

^ ' « {M |M Z ji).

A meet operation cuid a join operation may be defined

in ^ * as follows: N^A M » (M V N)**", and

10

H V JT « (N A,M) *

With these operations j ^ * is called the dual lattice

of ^ * Note that for a subspace M of E, M is a(E•,E)-

closed* Thus the dual lattice is a lattice of a(E*»E)-

closed subspaces of £*• If ^ : E -• E is a(E,E')-con-

tinuous, then the dued map * : £'-»>£* defined by the

identity,

< X, ••(x«)> « <*(x), x»>

is also o(E**E)-continuous [12]*

Lemma 1*6: Let ^ be a lattice of closed subspaces of

£ and j^* the dual lattice of closed subspaces of E*.

Then • e A l g ^ if and only if the dual map •• z Alg^,it'.

Proof: Suppose ^ e Alg V , then for each M e ,

• (M) 5 M . If N^ e*6*. vhere N e;^, then for n» e N^,

0 = <^(n),n»> » <n, •'(nM >

for every n c N. Thus •'(n*) e N so that

The converse follows by duality.

Proposition 1.7t Let be a lattice of o(E,E')-

closed subspaces of £ and jt* the dual lattice* Then

sC is a(E,EM-reflexive if and only if ^^ is a(E« ,E)-

reflexive*

Proof: Suppose that ^ is a(E,E*)-reflexive with

12

M» € Lat Alg^« and •* e Alg^«. According to

Lemma 1.6, 4 z Alg^ . Now for m e (M») and m* z H\

0 • <m, ••(m»)> « <*(m), m*>

so that •(m)€ (MM . Thus •[(M») J ^ (M») for every

^ E Algjf . Hence, (M») e Lat Alg^ *j( , which

implies M« » (M» ) ^ c J^*. Thus Lat Alg ,jf • S ;^' •

Proposition 1*8; If JC is reflexive, then

<5d » ^ Lat(A). A e Alg^

Proof: If A e Alg^^ , then <A> C. Alg^ which means

that Lat(A) « Lat<A> p Lat Alg ^ » ^ . Thus

^ e n Lat(A). A e Alg^

On the other hand, if M c ^^ Lat (A), then M z Lat (A)

A z Alg;C

for every A z AlgjC • This means A(M) C M for every

A € Alg 5^ which implies M c Lat Alg » ^ .

CHAPTER II

FINITE DIMENSIONAL RESULTS

Let (£,£•) be a dual pair, J^ a reflexive lattice

of o(£,E* )-closed subspaces and ui^ Alg ^ . For N a o

subset of ^ , define N as follows:

N* = {x e £ [•(x) = 0, • e N).

For V a subspace of £, define Vt as follows:

V+ = (• e /?U(x) = 0, X e V }.

Lemma 2.1: For each subspace V C E , V is a left ideal.

Proof: If • € V**" and 6 e ^ , then

e«(x) = e(^(x)) = 8(0) = 0

for each x E V, which implies 8^ z V*.

I^mma 2*2! If V e;^ , then V is a right ideal*

Proof; If • e V"*", e € and x e V, then

(x) = •(e(x)) * 0 since 8(x) z V for each x e V.

This implies that <8 z V"*".

Lemma 2.3: Suppose N is a right ideal of ^ , then

N* e ^ . That is, N* is ^-invariant.

Proof; Let x e N , 8 e and • e N* Since N is

13

a right ideal , ^8 c N which iutplies

•(e(x))= •e(x) » 0.

Therefore, 8(x) z N .

Lemma 2*U: If V is a subspace of E, V**" is an ideal if

and only if V*** e ^ .

Proof: Use the fact that V'*'•*'= V and the pre­

ceding lemmas.

Lemma 2.5: If V c;^ , then v'**e;d .

Proof; Apply Lemmas 2.2, 2.1, and 2.U,

An ideal N of ^ is said to be nilpotent if there

exists an integer r such that N^ « 0. An algebra ^ with

minimum condition on left ideals is said to be semisimple

with minimum condition on left ideals if it does not con­

tain a non-trivial nilpotent left ideal [9]* In such

cases, we will simply say that ^ is semisimple with

minimum condition.

A lattice is said to be atomic if every element is

a Join of atoms.

Proposition 2*6: If ^ has a non-trivial nilpotent

left ideal, then «C is not atomic.

Proof; Let N be a non-trivial nilpotent left ideal

lU

in * Suppose a is an atom of cC • Now Na z ^ ,

since for each • e ^ , •(Na) =« •N(a) e Na. The two

conditions Na a and Na e imply that either Na = 0

or Na * a* If Na » 0, then a c N ; on the other hand,

if Na « a, then since N is nilpotent, there exists an

integer n such that

0 = N' a » N°"" (Na) = N^"^a « ^^-^Hvia.) «...

= N^a = N(Na) = Na,

vhich implies that a ^ N . Consequently, €LL1 atoms are

contained in N*, which implies that ^ is not atomic.

Corollary 2*7: Suppose ^is an algebra with minimum

condition on left ideals. If «^ is atomic, then K is

semisimple with minimum condition*

Note that when E is finite dimensional, the algebra,

a. a: Alg#C , automaticeilly has minimum condition on both

left and right ideeJ.s.

Proposition 2*8: Suppose that £ is finite dimensional.

If U^ is semisimple with minimum condition, then «C is

atomic*

Proof: Every non-zero ^-module is a finite direct

sum of simple ^-submodules* Since ^ can be identified

vith the submodule lattice of ^ , then )C is atomic.

15

16

Combining Propositions 2*6 and 2*8 and using the

feu:t that a modular lattice of finite length is atomic

if and only if it is complemented, ve have:

Theorem 2*9: If E is finite dimensional, then ^is

semisimple vith minimum condition if and only if ^ is

complemented*

In the remainder of this chapter, E vill denote a

finite dimensional, complex, Hausdorff, locally convex

space* Also, A vill be used to denote a continuous linear

transformation from E to E*

Definition 2*10; A non-trivial reflexive lattice of

subspaces of £ is maximal reflexive provided whenever *

is a reflexive lattice of subspaces and elt^ ^ X * ^^^^

either it' 'X,^^ Jt* consists of all subspaces of E.

Theorem 2*11: The lattice of all reflexive lattices of

subspaces of a finite dimensional space £ has finite

length*

Proof; Suppose X^S: jf« S-• •SJ^n^*' * ° *"

ascending chain of reflexive lattices* Then we obtain

the following descending chain of algebras:

Alg#^l 5 Alg ;i 22 ..*pAlg ^ ^ • . . •

17

But since Alg {0> is a finite dimensional vector space,

it has finite length 1= dim Alg {0>. Consequently, the

lattice of all reflexive lattices must have finite length.

Corollary 2.12; The lattice of all reflexive lattices

of subspaces of a finite dimensional space £ satisfies

both the ascending chain condition and the descending

chain condition.

An element a of a lattice L is said to be meet-

(Join-) irreducible if whenever a = b A c ( « b V c ) ,

then a « b or a = c. Lattices which satisfy both chain

conditions have the following two properties.

Corollary 2.13: Every reflexive lattice of subspaces

can be expressed as a meet (Join) of meet-(Join-) irre­

ducible reflexive lattices.

Corollary 2.lU; Every reflexive lattice of subspaces is

contained in a maximal reflexive lattice of subspaces.

For a linear transformation A in E, there is a direct

decomposition of £ into gener6j.ized eigenspaces of A [U].

This means that there exist A-invariant subsx>aces E ,

i s 1,..., n, such that

n £ = 0 E .

i«l

The projection onto E. will be denoted by irj* The

projection operators n^ for i » I****, n, are poly­

nomials in A. For any subspace M e Lat (A), it is

known that M ^ E^ c Lat (A) for i = 1,..., n.

Proposition 2.15: Let A be a linear transformation in

n E with E s ( E. a decomposition of £ into generalized

i«l ^ n

eigenspaces. Then Lat (A) « ^ Lat (Aj^), where

18

i=l A » Aw.. 1 n

Proof; Suppose M e ^ Lat (A.) and m c M. Then i»l

m » m, • m^ •...••• m ^ with m. c E for i « 1,..*» n and

n A (m) = A(m, ) z M. Thus A(m) * I A(m^) c M, which 1 i»l

n implies ^^ Lat(A ) C. Lat(A). On the other hand, for

i=l

M e Lat (A), since M ^ E c Lat (A) for each i, then

ir.(m) e M ^ E for each m e M. Therefore,

A(ir^(m)) = A^(m) e M ^ E^.

Consequently, n

Lat (A)S ^ Lat (A ). i-1 ^

19

A linear transformation A is cflklled semisimple if

every A-invariant subspace has a complementcury A-invar­

iant subspace. It is nilpotent of index k provided

A » 0, but A^"^ + 0. Since every linear trimsformation

A in E can be written as a s\]m of a nilpotent trans­

formation and a semisimple one, each of vhich is a

polynomieil in A [U], then the folloving theorem may be

shovn.

Theorem 2*l6; If A is a linear transformation in E,

then Lat (A) « Lat (A ) ^ Lat (A^), where A « A • A ,

A is nilpotent, A is semisimple, and each are poly-n 8

nOTiials in A.

Proof: Note first that if A^(M) C M and A (M) 9 M ,

then (A • A )M C M. Thus n s ~

Lat(A^) f"^ Lat(Ag) CLat(A„* Ag) - Lat(A).

But if M € Lat (A) = Lat(A_ • A ), then A (M) C M and '» s n

A (M) C M since A and A are polynomials in A. This s — n s

means Lat(A) 5Lat(A^) ^^ Lat(A^).

Now the question of interest becomes that of

describing Lat(A) where A is either nilpotent or semi-

simple.

In a lattice ^ an element M in ^ is said to

be conqpletely meet-irreducible if whenever

20

M = A <M I M 6 olS > o zj^, a a

then M « MQ for some o . The following is a well-

known consequence of this definition.

Theorem; A lattice ^ of subspewjes of E is completely

meet-irreducible provided that there exists a subspace

M i jt with the property that any lattice of subspaces

which properly contains iC must contain M.

Note that a maximal reflexive lattice of subspaces is

completely meet-irreducible in the lattice of all re­

flexive lattices of subspaces of E.

Theorem 2.1?; If A is nilpotent of index k and Alg Lat(A)

consists of all polynomials in A, then Lat(A) is completely

meet-irreducible in the lattice of all reflexive lattices

of subspaces of E, Moreover, when k = 2, Lat(A) is max­

imal reflexive.

Proof; Suppose JC is reflexive and J C ^ Lat (A).

Note that Algdif C Alg Lat (A) and A i Alg X > Choose

x z ker A^^ ker A. It will be shown that < x > c X while

< X > i Lat(A). If B e Alg jC • ^^n since B is a poly­

nomial in A, B has the following form;

k-1 B « y^I • w^A +...• ^y^^-^A • vhere w^ e C, i«l,...,k-l.

Since Alg^ contains scaler multiples of the identity.

21 vithout loss of generality, ve may assume y » 0. By

considering the elements B, B^,***, B^"l, all in A l g ^ ,

it is seen that there exist a € C, i « 1,..*, k-1 such

k-l ^ that I a.B « y A* Thus, y- « 0, since, othervise,

i«l " 1

A e Algj^ . Consequently, B(x) e < x > for each

B e Alg^ , imich means <x> e X. •

For k » 2 if ^ O Lat (A), then since

Alg i^CAlg Lat (A) « {XI • yA |X,y z C},

either Alg jt » Alg Lat (A), in vhich case cl(« Lat (A);

or Alg;^ consists only of sceaer multiples of the identity,

in % iich case ^ consists of all subspaces of E*

Proposition 2*l8; If P is an idenpotent linear trans­

formation in £, then Alg Lat(P) is equal

{ XI • yP I X,M e « }*

Proof; Since for m e im P, there exists n e £

such that P(n) « m, then P(m) « P(P(n)) * P(n) » m.

Hence, M e Lat(P) for each subspace M Im P. Also every

subspace of ker P is in Lat (P).

For A e Alg Lat (P) and m e Im P, then A(m) " X m m

for some X ^ e C * I f p e I m P vith p and m linearly in­

dependent, since there exist X e C and X e C such that P p4m

A(p) » X p and A/^j^\(p+m) « ^fTy»-B\^P^^» ^^^^ ^ linearity and independence, X « X^^ " ^ . Thus there exists X c C

p p^m ™

such that A(m) « Xm for each m e Im P* 5y a similar

argument, it may be seen that there exists Y c C such

that A(m) « -ym for every m e ker P*

Since £ » ker P Im P, for each x c E, then there

exists m c ker P and n c Im P such that x « m ••• n* Nov

n * P(q) for some q, and

A(x) « A(m) • A(n) » ym •»• Xn •»• ^[m • n • (X-Y)n]

« Y(m • n) + (X - Y)P(q)

« Y(m + n) • (X - Y) P(P(q))

= Y(m • n) -»• (X - Y) [P(m) • P(n)]

= Y(m • n) • (X - Y) P(m • n)*

Thus A » YI • (X - Y) P.

Theorem 2*19: If a linear transformation P in E, P O and Pfl^

is idempotent, then Lat(P) is maximcd reflexive.

Proof: Since Alg Lat (P) » {XI + YP |X,Y C C },

then as in Theorem 2.17, Lat (P) is maximal reflexive.

Lemma 2*20; If A is nilpotent of index k > 2, then

Lat (A^) O Lat (A), for each i with 1 £ i <_ k.

Proof: Choose x e E such that A^(x) + 0. Then

i i 2 i < x, A (x), (A ) (x), ...> is invaricuit under A* but not

under A*

Theorem 2.21; If Lat (A) is maximal reflexive, then A

22

is nilpotent if and only if A^ » 0*

Proof* Since Lat(A) is maximal reflexive, then

Lat(A^*^) « Lat (A); but by Lemma 2.20, k must equal 2.

Note that Theorem 2.19 and Theorem 2*21 are true

for infinite dimensions.

An A-invariant subspace M of E is said to be A-in-

decomposable if and only if there do not exist non-triviia

A-invariant subspcu:es M. and M such that M » M- ® M .

Every A-invariant subspace has a direct sum decoiq>osition

into A-indeconposables [U].

Theorem 2*22: If J^ is maximjJ. reflexive, then either

there exists em idempotent linecur transformation P in E

such that o C * Lat (P) or there exists a linear trams-

formation B in E which is nilpotent of index 2 such that

^ = Lat (B).

Proof: Let A z Algit and suppose that A is not a

scaler multiple of the identity. Since ^ is meucimal re-n

flexive, then Lat(A) = *t • Let E » ® E be a decompos-

i=l ^

ition of E into generalized eigenspeu:es. Since w , the

projection on Ej, is a ^^olynomial in A and if w* f I for

some J, then Lat (vj) « Lat(A). In case n " 1, then w^ I

n and E » O F. , where F z Lat (A) and F is A-indecomposable

ial i 1 i

23

for each i* Then E » ker (A - Xi)n, where x Is the

eigenvalue associated with E. That is, A - xl is nil-

potent of index n > 1* If B = (A - XI)" vhere m > n/2,

then Lat(A) • Lat(B) and B^ » 0*

Theorem 2*23: If A is semisimple, then Lat(A) is a finite

intersection of maximal reflexive lattices*

Proof: n

Let £ = ® E be a decomposition of E i«l ^

2U

into

generalized eigenspaces. Let y » (t - Xj^)(t-X2)**.(t-XQ)

be the minimum polynomiea of A* Then A|E. - X.I * 0 since

t - X^ is the minimum polynomial of A|E. for each i. Nov

considering {A > as defined in Proposition 2.15, ve have

t^.I on E^ - ^^ "^^9 i = l,...,n. Thus

0 elsewhere

Lat (A. ) s Lat (ir. ) which is maximcd reflexive* But n

Lat(A) - ^ Lat(A. ) implies that Lat(A) is an intersection 1=1 ^

of maximal reflexives.

For a nilpotent transformation A of index k there is

an ascending sequence as follows;

0 C ker A C ker A^ C ... C ker A^"^ C ker A^ » E.

Also for X e E, if A'*(x) + 0, then x, A(x), A^(x),*.*, A'*(X)

are linearly independent. For more information on nil-

potent transformations see [3], [U], and [8]*

Theorem 2.2k: If A is nilpotent of index k » dim E, then ^5

Lat (A) is a complete chain*

Proof: See [2] for proof*

If A is nilpotent of index k, an A-invariant subspace

M of E, of dimension m, is called cyclic with respect to

A if there is an element z e M such that z, A(z),..., A"'"^(Z)

form a basis of M. The element z is said to generate the

cyclic subspace M. The two following well-known theorems

[8] describe the nature of nilpotent linear transformations.

Theorem: Let A be nilpotent of index k. There exists a

unique set of integers {r. | i = 0,..., n} with

n k « r^ > r, > ... > r^ and I r. - dim E such that 0 - 1 - - n ^^Q i

n £ » 0 E , where E. is cyclic with respect to A and

i-O i ^

dim E « r for i = 0,..., n.

The integers r ^ , . . . , r are ccLLled the invariants of A. O n

Theorem: Two nilpotent linear transformations are similar

i f emd only i f they have the same inveiricmts*

These tvo theorems v i l l prove useful in describing the

l a t t i c e of invaricmt subspaces for a nilpotent linear

trans forsiation •

In the discussion that follovs rQ « k and r « r.

Lemma 2*25: Let A be nilpotent of index k < dim E* If

k » r or k = r+1, then Alg Lat (A) consists of all poly­

nomials in A*

Proof; Case I ( k « r) :

n I«t E « ® E. be a decomposition of E into a direct

i«0 ^

sum of cyclic subspaces* Since k « r there exist x c EQ

and y € E^, vith {x, A(x), ***, A^"^(x), y, A(y),**., A^-^(y))

linearly independent, E « <x, A(x),..*, A^'"^(x)> and

E^ " <y, A(y), *.*, A^"^(y)> .

For B e Alg Lat (A), ve have the folloving:

1*) B(x) * p (A)(x), vhere p is a polynomial of degree

less than k;

2.) B(y) « p (A)(y), vhere p is a polynomial of degree

less than k; and

3.) B(x-^y) « p . (A)(x+y), vhere p is a polynomial o' x^y x^y

degree less than k.

By the linearity of B and the independence of the set

{ X, A(x),..*, A^"^(x), y, A(y). ..., A^"^(y)} , it is seen that p • P ^ " P..

X x- y y

For any z c EQ there exists m < k such that A"^(Z) • 0

vhile A"*"^(Z) + 0* Then

26

<z, A(z),**., A * (z)> e Lat(A) and

{z,A(z),***, A (z),y, A(y), ***, A^"^(y)} is linearly

independent* Consequently, B(z) « q^(A)(z), i^ere q

is a polynomial vhich agrees vith p^ » p , term by term, • X

through the term of degree (m-l)* Since A (z) « 0

for 1 I m , then B(z) « Py(A)(z)*

Also for z e E^ vith i f 0, there exists m <, k such

that A"*"^(Z) + 0 but A°(z) » 0. Since

< z, A(Z),..*, A"*"^(Z) > is in Lat (A) and

{Z,A(Z),**., A ° " ^ ( Z ) , x,A(x),..., A^"^(x)} is linearly

independent, by an argument similar to the one above, it

is seen that B(z) » p (A)(z) = p (A) (z)* X y

Therefore, for each B c Alg Lat (A), vith k « r, there

exists a polynomicLl p such that B - p(A). Case II ( k = r+1 ) :

Let X be a generator of £ and y a generator of E.,

As in Case I, it may be shovn that if B(x) = p^(A)(x) for

some polynomial p„, then B(y) = p^(A)(y). In fact, for * X

any z c E vith i + 0, B(z) « p^{A){z). It remains to k-1

consider the case vhere z c £ = <x, A(x), ..., A (x)> * o

If A'*(Z) = 0 and A (z) + 0 with n < r, an argument like

the one in Case I shows that B(z) = p (A)(z) « p^(A)(z)* y *

Now if A^"^(z) + 0, then <z, A(z), ..., A^"^(z)> = EQ.

Let p be a polynomial such that B(z) • p (A)(z). Con-z z

27

slderlng B(z) and B(y) and using the linearity of B and

the independence of the set

(z, A(z), ***, A^-^(z), y, A(y), .... A^^^iy)} ,

it is seen that p^ and p^ agree on the first (k-1) terms.

Now consider the folloving:

B(x+z) « XQ(X*Z) • X^A(x*z)+**.*Xj^_2A^-2(x*z)^«A^"^(x*2),

B(x) « p^(A)(x) » XQX • X^A(x)f.*.^Xj^^^A^"^(x), and

28

k-2, k-1 B(z) « XQZ + X^A(Z) *...'*'\^^A^~''{Z) * oA (z).

By linearity, «A * (x'«-z) « ViA^"^(x) • <'A^"^(z)* If

A "" (x) and A^"^(z) are linearly independent, then

5 » ^^^^ » a and B(z) » p (A)(z). On the other hand,

k—1 k—1 suppose XA (X) « A (z) for some X e C* If X « -i, then

^k-1 " ^^^ B(z) « p„(A)(z). There is no loss of gen­

erality in assuming that X « -1, for if not, then let

z* » - z, . Then A^"^(x) = - A^"^(z») so that X

B(zM a P3^(A)(z»), but B(z) = -XB(z') - - p (A)/-z, » r)

P, (A)(z)* Consequently, there exists a polynomial p such

that B « p(A)*

Theorem 2*26: With A as in Lemma 2*25, Lat(A) is com­

pletely meet-irreducible in the lattice of all reflexive

lattices of subspaces of E* When k • 2, Lat(A) is max-

imal reflexive* ^

Proof: Apply Theorem 2.17*

Theorem 2*27: Suppose A is nilpotent of index k < dim E*

If r • 1 and k > 2, then Lat (A) is an intersection of

maximal reflexives*

Proof; Suppose x is a generator of the cyclic sub-

space EQ* Choose X* £ £• such that < A*"'^(x), x'> • 1

and < A^(x), x' >« 0 for i < k-1* Let

A * {A^(x)€)A'J'(x») |i>.l, i * J » k }.

It will be shown that Lat ^ « Lat (A)* For M e Lat (A)

and k-l< m c M , m « X x + X A(x) •.*.• j A* («) • y» vhere

y e S £ • Note that since r » 1, A*'(y) • 0 for every i»l ^

J > 1* Now

A ^ ( X ) O A J ' ( X » ) (m) « [< XQX, A'^*(XM> •

< X A(x), A"* (x')> •...• <X A " (x),A''*(x')>lA^(x) 1 K—X

» X A (x)*

Since A(E) * A(EQ) » <A(x),..., A^"^(x)> there exists

z ^ z such that A(M) » A^(E). If i >. E then A^(x) z M N and A^(x) ® AJ'(xM(m) c M. On the other hand, if i < c,

then since

A«-l(m) - XQA^-^(X) •X^A^(x) •••••\.(c.i)^*"'^<*^ ^ "

and A^(x), .**, A^"^(X) C M . we have X A^"^(x) e M* But

since A^" (x) i M, then X « o* Suppose that for each 1 30

0

such that i < n < e , X « 0 for all J <, i. Nov for i • n,

since X « 0 for all J < n-1, then

A^-"(m) » ViA^"^(x)+ X^A^(x)^**** \^{t^^)^'^M e M*

But this implies that ^n,iA^"^(*) **, vhich, in turn,

iomlles X , * 0* Thus for all i such that i <e , X. . * 0* n-1 1-1

Therefore, in all cases, A^(x) A '(x« )(M) O M so that

Lat(A)^Lat ^ *

Next let M e Lat and m e M vith k-1 JSL

m « XQX • X^ A(x) •***• \.iA* -"-(x) • y, vhere y c © E^.

Since for each A^(x)0A^'(x) c ,

A^(x)©A*^'(x»)(m) » X^^^A^x),

vhich is in M, then either A (x ) c M or X « 0 for each 1-1

i > 1* Since for each i vith if. e-1, A (x) | M, then

X._ « 0. Consequently,

e €*•! k-1^ ACn) « X A (x) •X A (x) •...• X A (x).

e-1 € k-2 g c-H k-1

But A (x), A (x),..., A (x) c M. Therefore,

Lat 4 CLat(A).

Hence, Lat (A) « Lat « ^ Lat(B). Since every BeO.

element B of ^ is nilpotent of index 2 and dim E > 2,

Lat(B) is maximal reflexive*

31

The final case to consider for a nilpotent linear

transformation of index k < dim E is the one in vhich

r > 1 and k > rt-l* Before this case is examined, several

preliminary lemmas vill be proved*

In the lemmas that follow suppose that J^ is re­

flexive cmd U^ Alg X * Let U be invertible and de­

fine ;^ « {u" M I M €«C}*

Lemaa 2*8: Alg i ^ « XT^Oi "•

Proof: For M c , u"^Au[u"^] « U"^^(M) u"^*

Thus U"^ 4 U 5 Alg ^ . For B e Alg ^ ", since

B[U"^M] CU"^M for every M c^^, then

UBU"^M UU"^M » M*

Thus UBU"^ € O, • vhich means that there exists AzQ^ such

that UBU"" « A. Therefore, B « U"^AU.

Lemma 2*29: I^ ;t 3jt t ^«« /, • X, .

Proof: Note that the map vhich sends A to U"^AU is

an injection of Alg to Alg iC . Therefore,

dim A l g ^ ^ dim Alg j^^. On the other hand,

Alg id ^ Alg JL . Therefore, Alg « Alg X^ , vhich

U ^ U > V implies ^ ^ lAt Alg ;C " ^ ^ AlgJC^ « ^ *

LeiMia 2*30: If B e Alg Lat (A) and B • U"^A U for some

invertible U, then Lat(B) « Lat(A).

Proof; First it vill be shovn that Lat(B) • / ^ , ^

vhere ^ « Ut(A), Since B e Alg ^ " , then

Lat (B)OLat Alg^P^^O ^ " .

For M £ Lat (B), since B(M) « U-^AUM 5 M , then AUM 5 UM

so that UM £ Lat (A)* Because M « U"^(UM), ve see that

M e ^ .

Since B £ Alg Lat (A), then

Lat(B) « ^ " - (Lat(A))"£Lat (A).

^y Lemma 2*29 this means that Lat(B) a Lat(A)*

Theorem 2*31: If A is nilpotent of index k < dim E and

r > 1, k > r -l, then Lat (A) is completely meet-irreducible

in the lattice of reflexive lattices of subspaces of E*

Proof; Choose x a generator of the cyclic subspace

EQ. If ^ P Lat (A) and Xis reflexive, then

A l g ^ ^ Alg Lat (A) and A 4 Alg ;^ .

For B £ Alg;C CAlg Lat (A), B(x) « p (A)(x), where

~ X

p is a polynomial. For each y £ E. with i ^ 0, a method

identical to the one employed in Lemma 2.25 shows that

B(y) « p^(A)(y). Now for z e E , since there exists a polynomial p

0 z such that B(z) » p (A)(z), as in Lemma 2.25, it may be

z shown that p^ and p. agree, term by term, through the term X z

of degree (r-l). After this point they may fail to agree*

Thus B » XQI • X.A • N, where for each y £ E there exists

y« £ <y,A(y),..*, A " (y)> such that NA (y) • A '* (y*)

for all t s 0,***, k* Suppose X^^ 0; without loss of

generality, we may assume X « 0 and X » 1. Thus, 0 1

B « A ••• N. Note that both A and B are nilpotent of index

k. For each x £ E, B**^(x) « A**^(x) + A*'*' (xM for some

k—1 x* £ <x, A(x),***, A (x)> . To see this, suppose first

that t = 1. Now

B^(x) » B(A(x)+A^(x^)) » A(A(x))+ A(A^(xj^))*N(A(x))-i-N(A^(x^))

=A^(x) • A3(X ) • A3(X^) • A^(x ) 1 2 3

»A^(x) • A3(X»).

Where x , x £ <x,A(x),..., A (x)> , and

X £ <x,,Ax ..*., A^"^(x )> O <x,..., A (x)> , 3 - ^ 1 1 ""

k—1 whence x* £ <x, A(x),..*, A (x)> .

33

t t ^*^ Suppose for t, we have B (x) » A (x) • A (x*) with

x» e <x, A(x),..., A ""'-(x)> . Then

t+l B^*^(x) = B(B^(x)) = B(A^(x) • A (x^))

= A(A^(x)) + N(A^(x)) * A(A^*\x ))* N(A**^(x^))

= A ' ' \ X ) * A^*2(x^) > A^*^(x^) * A^*\x^)

aA^*^(x)*A^*V). k-1, .

where x., x e <x A(x),.*., A (x)> and

X3 £ < x^,..., A*'"^(x^)> S ^^. A(x),.*., A " (x)> .

Thus x» € < X,***, A '' (x)> . ^^

Let k » rQ ^ r^ > r^ > *.. r^ denote the invariants

n

for A and E » ® ^ be a cyclic decomposition determined i«0 "

by these invariants. Since B £ Alg Lat (A), then B(E ) C £ ,

for each i. Also, if x is an A-cyclic generator of E , that

r*-1 is, E^ = <x, A(x) A 1 (x) > , then since for some

X' € E', B^'i'^x) « A ' I - ^ X ) • A'i^^Cx') - A'^I'^X) 4 0,

r • —1 then <x, B(x),..., B ^ (x)> » E^. That is E^ is B-cyclic.

Therefore, B and A have the same invariants; and, con­

sequently, are similar. Then by Lemma 2.30, Lat(B) = Lat(A)*

Thus A £ Alg Lat(A) = Alg Lat(B) c Alg olf , which is a

contreuliction. Therefore, X « 0. 1

2 Now choose z e ker A " ker A. Since < z > e Lat(B)

for each B z Alg Jt , but <z > ^ Lat (A), then Lat(A) is

completely meet-irreducible.

We see that ccoipletely meet- irreducible lattices play

a very important role in the description of inveuriant

subspace lattices in a finite dimensional space. Since

maximal reflexives are cooipletely meet-irreducible, then

all id«npotent operators give rise to completely meet-

irreducible reflexive lattices. This holds for spaces of

infinite, as well as finite, dimension. The invaricmt sub-

space lattice of a semisimple operator, it turns out, 35

is an intersection of maximal reflexive lattices. In

all cfiises nilpotent operators give rise to lattices

which are either completely meet-irreducible themselves

or an intersection of maximal reflexives.

CHAPTER III

WEAKLY-REFLEXIVE ; MACKEY-REFLEXIVE

Theorem 3.1; Let (E,E») be a dual pair. The weakly-

reflexive sublattices of closed subspaces, and the

Mackey-reflexive sublattices of closed subspaces coin­

cide.

Proof; A map from E to E is a(E,E' )-continuous if

and only if it is m(E,E*)-continuous [12],

The issue of reflexive lattices of subspaces in

Hilbert space has long been of interest ( [5] and [6]).

In most of this work heavy reliance has been placed on

properties involving use of the inner product. The above

theorem points out that in Hilbert spaces, or, more

generally, in Banach spaces, since the norm topology and

the Mackey topology coincide, questions of reflexivity may

be resolved by using the weak-topology only.

Suppose T is any topology which is compatible with

the duality. Since every T-continuous map is weakly-con­

tinuous [12], then Alg ^ O A l g 0^ for every lattice

^ of weeikly-closed subspaces of E. Therefore, whenever

T is compatible with the duality, it is always the case

36

that Lat Alg ^ I^ Lat Ala J^ m , D ^j^ - ^ ^ 8 g • To say that Alg JC

is weakly dense in Alg Z means that for each • £ Alg ai

there exists a net {8 } with 8 £Alg ^ such that

{8^} converges weakly to ^ ; that is, for every x £ E

and X* £ E«, lim< 8 (x), x* > « <#(x), x* >* Y '

Proposition 3*2: Suppose that jC> is a(E,E»)-reflexive

and that T is a topology on E which is compatible with

the duality. If Alg ^ is weakly dense in Alg ^ , T O

then >C is T-reflexive.

Proof: Since ^ Lat Alg^ it , it suffices to show

that Lat Alg^ O Lat Alg^ ^ ^it . If M £ Lat Alg ji and ^ £ Alg^ , then there exists a net {8 > ^ with

^ Y Y £jt

8 £ Alg 'J!^ such that {8 > converges weakly to •. But

since 8 (M) C.M for each Y ^UL % then •(M) C M . This

implies M £ Lat Alg X. • 0

Corollary 3.3: Suppose that •£. is a lattice of o(E,E*)-

closed subspaces, and T is a topology on E which is ccxn-

patible with the duality. If Alg ;6 is weakly dense in

Alg X, . then Lat Alg ^ = Lat Alg i^ . C T O

An important question which naturally arises is

whether the reflexive sublattices of closed subspaces coin­

cide in all topologies compatible with the duality. It is

37

38

apparent that if T is any topology compatible with the

duality and d(^ is a lattice which is T-reflexive, then,

since pC = Lat Alg aC ^ Lat Alg ^2^ , Jt- is also T 0

o(E,E»)-reflexive. On the other hand if the lattice is

a(E,E•)-reflexive, it is not necessarily T-reflexive as

the following example illustrates:

Consider the space S of bounded linear operators on

a separable real Hilbert space H with orthonormal basis

(e ) . Let HJ* denote the trace class and ^ t h e Schmidt ^ i«l

class on H [13]. Note that Jt f'Ji^'^ ^ is a norm ideal

in ?S . For A z tSi ^ the Schmidt norm of A, denoted \A\a ,

1/2 is equal [ I I l(Ae ,e )! 1

J i '^ ^

Define ^\ ^ ^ d -• IR by

•(B,T) - tr(TB) = I (TBe.,e ). It may be observed that

• defines a duality between flS and ,^ .

Let A = e. 0 e^ for i = 1, ...,~ . Note that A^ e ^

for each i. Define a seminorra P, where P : tt> - R ,

2 1/2 withP(B)= I A B L =[II(Be .e,)! ] .

^ M J

Let 6 be the coarsest topology on U''such that each

T £ ^ is continuous and p is continuous. Thus we see

that a( d5», 5 ) C 6. It will be shown that B S m( ® . ^ )•

Consider S « { B | p(B) < z) for some £ > 0. Since S

is absolutely convex and absorbent, if S is weakly closed,

then S is a barrel. Proving that S is a barrel will

suffice to show that 8C:m((&, SZ ).

Suppose {B } is a net such that { B } -• B in a

<'( V* 3i ) with B £ S f or each a . Then {tr TB } a o

converges to tr TB for each T £ . Now if

tr(AjB„) - I (»aje^. e^) = X^ ^^ for each 1 and

0 tr AtB = XJ,. Therefore, X - X for each i; and,

"1 il ail il

39

t i y . |A^,J^

fact that

consequently, U^^, I •*'l^iil • "^^^ together with the

2 1/2 [ Zl^aiil J 1. ^ '"or *ftch a implies

that 2 1/2

[ I I \ J ] l e . i ii

2 1/2 But p(B) « [ Z U^il ^ 1 ^ • Consequently,

i ^^

S is weakly-closed. Thus 0 ^MtS %(H )*

Now it will be shown that the mapping J : t-*- © defined

by J(B) s B , where B denotes the adjoint of B, is a((S,tlt)-

continuous but not 8-continuous.

First to see that the mapping J is a((»',2Zr)-cOntinuous,

suppose that {A } is a net in 9t which converges to

0 in a((S( • ); thus, for each T £ 55 , tr(tA ) con-a

verges to 0. Since T £ ji implies that T* £5< , then

tr(TA^*) « tr(A^T )* = tr(A^T*) » tr(T*A ) converges

to 0. Consequently, (A^*) converges to 0 in a( A ,ift ).

Thus, J is a( [!pt,CA)-continuous. to

On the other hand, consider the sequence { A } ,

""m l

where A - e (5)e,. Note that A = e, (g)e for each m.

Since for each T £ w^ , tr(TA ) = (e ,T*e,), then we see m m l

that {A } converges to 0 in a(O0',2'); and, hence, m

{A } converges to 0 in aidb** 5«)» Now it will be m

shown that {A } converges to 0 in B while { A } does m m

not converge to 0 in B. This will then insure that J is

not B-continuous. First p(A„) = |A A_ | ' I I l(A„ej.e^)| ^)

1)0

2 .1 /2

• I l |(e © e <e ).e ) | ¥ ^ ^ V m 1 J 1

« 1/2 « [ I Kej.e )(e^.e^)l ^]

J

• {

= | (e . e . ) I m 1

•0 if m + 1

,1 if m « 1

Therefore, {Aj } converges to 0 in B . However,

P ( A ' ) . [ I l(A^%,.e,)l¥^^-[n(ei®e,(e).e^)|2) ™ J J J

1/2

2,1/2 *»1

= 1.

vhich implies that {A } does not converge to 0 in B* m

Thus J is not B-continuous.

Lemma 3*U; If A s (^ , then A and A* are independent if

and only if A 4 *** A .

Proof: Suppose A « XA , then

(A(x),y) » (x,A*(y)) - 1 (x,A(y)) « X(A*(x),y) » X(x,A(y))* X

Therefore, 1 « X vhich implies that X « + 1* X

Lemma 3.5: Let A, B E 69* vith A + + A andB + + B ,

There exists C £ ^ vith C + • C* such that A, A*, C, C*

are linearly independent and 6« B , C, C are linearly

independent.

Proof: Choose S and K not in the linear span of

A, A*, B , B vith S = S and K = -K . Set C - (S-K) . 2

Note that S » C • C and K « C - C.

Assume that C » oA '•' BA for some scalers a and B • «

not both zero. Then S = (o • B)(A -•' A ) which is a con-

tradiction since S is not in the linear span of A and A .

Thus A, A , C are linearly independent.

Next suppose that C = oA • BA* • yC for scalers

U2

,B,Y vith Y 4 0* Since

o . 6 * 1 * O - - A - -^A • - C Y Y Y ,

it follovs that

C s - j S A - ^ A -i-l^C. 7 Y Y

Therefore, a = - ^ , B * - ^ and Y» 1,, vhich implies that Y Y Y

Y« • 1 and B - OY • In case Y* 1» then K « a(A-A ) which

is a contradiction since K is not in the lineeur span of A

and A . Considering the alternative, Y^ -1, shows that

S = a(A + A ) which is another contradiction. Therefore,

A, A , C, C are independent. A similar argument shows

that B, B , C, C are independent.

Let T : ^ — ^ d B * be ai^t 3* )-continuous. Con­

sider the following property for T;

For each B € Sw' there exist scalers

(3*1) a and Y„ svich that T(B) ^ a B + Y B . g B DO

Proposition 3.6; Suppose A £ 4& with A + + A*, T i^^OM

is a ( ^ ^S^ )-continuous, and T has the property described

by (3*1)* If T(A) « oA •»• YA* for scalers a and Y . then

T(B) « aB • YB for every B £ fii* .

Proof; Suppose first that A, A , B, B are linearly

h3

independent. Then a^ - a^^g « o^ and Y^ = Y^^^ - YfiJ

and it follovs ftrom Lemma 3*5 that the proposition is

true for all B vith B 4 • B*.

Nov suppose B = + B and let { A^} denote a sequence

of operators such that {A } converges to B in a ( ^ , 5 J )

and A^ f • ^n ^ ° ' ^^^^ ^* ^^^^^ " ^ o(»'. 3 )-con-

tinuous, then T(B) « lim T(A„) = lim (oA •»• yA *)« OB+YB*. n ^ n n ' n ' ' *

Corollary 3*7; If T satisfies the property described by

(3*1) and T is B-continuous, then y " 0 and T(B) « oB

for every B z 6B* .

Proof: If Y 4 Of ^hen J = Y~ T - OY"^I is B-continuous

since both T and I are B-continuous.

Let oG be the lattice of all self-conjugate closed

subspaces of ^ . Note that e£ is a{dB ^ V* )-reflexive,

since Lat Alg 00 s«C ; however, ^ is not B-reflexive. o

If A £ 28# define L^ »{ oA • BA* | a,B £ « }. It is

readily seen that od is generated ^y (L. | A c 1 ^ > *

If T £ Alg A d • then T ( L J <=• L. so that T(A) =0 A • BA*

for some a,B £ F. Thus T(A) = oA, by Corollary 3.7, which

implies that Algg OL/ consists exactly of scaler multiples

of the identity. Hence, Lat Alg id « (fl so that t& B

is not B-reflexive.

CHAPTER IV

COMPLETE CHAINS

It is well-known that complete chains of closed

subspaces in Hilbert space are reflexive [ll]. The

proof that follovs generalizes this resiat to an arbitrary

vector space E for vhich there is a duality.

I«t ^ be a complete lattice of closed subspaces of

E. For each closed subspaces M, define

M_ » V [L I L e od » L <ZM )

and M^ « A (L I L E ^ , M C L }.

Lemma U*l: Let ^ be a complete chain of closed subspaces

and M £ jf . If f E M and e' E (M^J^, then f ® e ' ( C ) ^ C

for every Q z 0 •

Proof; For M, L E , either M L or L M . Con­

sequently, f e*(L) L whenever f E M and e* E (M_) .

Lemma U*2; Let X^ \>e a ccmplete lattice of closed sub-

spetces with {0} , E £ X . For each closed subspace

M ^ , there exists L £ C such that L M and M ^ L .

Proof; Since M ^ sC » then M^ is not equal M_. If M^

covers M^, then M^ is the desired subspace. On the other

hand, if M^ does not cover M^, then choose L £ ^ such

that M^c L c M^* Note that L + M since

L C M^; and also M L^, since M C L^ implies that

k5

ii^SKS !*•

Theorem U.3: If (£,£•) is a dual pair and ^ is a complete

chain of closed subspaces of £ with {0> , E £ /^ , then

/^ is reflexive*

Proof: If M I , by Lerama U.2, there exists LE ^

such that L ^ M and M ^ L _ . Consequently, L^^M^ and

M^^(L_) , since L and M are closed subspaces. Now it is

possible to choose f E L x M with t f 0 and e* £ (L_) N M

with e* 4 0» By Lemma U.l, f®e*(C)^C for each C Ejf.

, 1 On the other hand, since e' M , there exists x E M such

that < x,e' > 4 0* Consequently, f0e*(x) M, otherwise

f £ M*

CHAPTKR V

A CLASS OF REFLEXIVE LATTICES

K* J* Harrison [7] has described a rather large

class of reflexive lattices of .subspaces in Hilbert space.

Included in this class are fini- .e, distributive lattices

and complete, atomic Boolecui al/ ebras of subspaces. With

the techniques of duality, these results may be generalized

to an arbitrary dual pair, (£,£').

In a lattice K0 of closed subspaces of E, a subspace

M £ jf is said to be completely Join-irreducible if when­

ever M = V { M I M e X . ) , then M = M for some o .

a a o

Completely meet-irreducible lattices are defined dually.

A lattice is said to be infinitely meet-distributive if it is Join- complete and

M A ( V M ) = V ( M A M )

holds for every subset {M^}^ zJL^^ ** ® lattice. An in­

finitely Join-distributive lattice is defined dually [lU].

In the following fl- denotes the set of all completely

Join-irreducibles in ©^ . For each closed subspace N,

p(N) = V (M I M £ <)^ , N ^ M >.

46

U7

Theorem 5.1* Let be a complete lattice of closed

subspaces of £ with {0} and E contained in JL . If, in

addition,^ has the following properties;

i.) ^ is infinitely meet-distributive,

ii*) each non-zero element of od is a Join of com­

pletely Join-irreducibles;

then d^ is T-reflexive, where T is any topology com­

patible with the duality.

The proof of the theorem will be preceded by several

lemmas.

Lemma 5.2; If N E o^ and <J is infinitely meet-distributive,

then N ^ p ( N ) .

Proof; Suppose N ^ p(N), then

N = NAP(N) = N A ( V { M | M E ^ , N ^ M } )

= V {N A M I M E^d , N ^ M >

^ V { W | W E ^ . W < : : N }

<=:N.

Consequently, M .

Lemma 5*3; Let be infinitely meet-distributive. If

N E (L and M E one and only one of the inclusion re­

lations N 9 M and M p(N) hold.

Proof: From the definition of p(N), it is clear that

U8 either N c M or M 5 p(N). If both relations hold,

then N C p(N), vhich is imposs:ble by Lemma 5*2.

Consider the folloving set d of operators:

d^ {Xj^OXjj* I N £ ^ , Xjj E N. Xj ' £ (p(N))^ } ,

vhere Xj,0Xjj'(x) = < x, x^> x . Note that the operators

in d cure continuous in all topologies compatible vith

the duality.

Lemna 5*U: If a^ is infinitely meet-distributive, then

;deLat^* Proof; For M E ^ , either N ^ M or M ^ p ( N ) for

each N £ O' . Thus x„CE)x '(M)^M for x„<g)x »£ ^ . O N N — N N -^

Lenma 5.5; Suppose «X is infinitely meet-distributive.

If M £ IjAtd and N zOr- , then exactly one of the in­

clusion relations N ^ M and M ^ p(N) is true.

Proof; Since dd Is infinitely meet-distributive,

at most one of the Inclusion relations holds. Assuming

M C p ( N ) inqplies that M"i[p(N)] so there exists

X* £ [p(N)] such that x» i M . Consequently, for some

m £ M, <mo,x'> 4 0» ^^ ^^^ arbitrary n £ N, ve have

n€)x* £ and n^xMrn^) » <m^, x»> n £ M since M £ Lat^*

Hovever, this implies N M.

l^Bna, 5*6; Let jd be infinitely meet-distributive and

each non-zero element of dC be a Join of completely Join-

irreducibles* Suppose M is a closed subspace and

V « V {N £ I N C M ). Then

{J I J e ^ , J ^ M } » {J I J£ Or , J^V> *

Proof: First note that for N £ CL , ve have N V

if and only if N M* Consequently,

{J I J E ^ . J ^ V } ^ { J | J eQ^ . J ^ M } *

Also for J £ , J V implies J M so that

{J I J £ ^ , J ^ M } ^ { J | J e ^ . J ^ V > .

Leamia 5*7: Let be infinitely meet-distributive and

each non-zero element of «C be the Join of completely

Join-irreducibles* If V E , then

V « A {p(J) I J e Q. , J ^ V } *

Proof: From the definition of p(J) ve see that

V ^ A{P(J) I J € ^ , J ^ V }* Let

W « A{ p(J) I J £ , J ^ V ) , and

suppose that N E , N ^ W and N ^ V . Since W ^ ( N ) ,

it follovs that N p(N), imich is impossible. Thus if

H c Q^ and N W, then it must be that N 5 V* Nov ve

have W « V ( K I K £ , K O W><= V, vhich implies

V . A { P(J) I J c ^ » J ^ V >•

The results of the preceeoing lemmas vill nov be

h9

liXAS ^£SH LI3RAR

50 used to prove Theorem 5*1.

Proof of Theorem ^.1; Suppose that M £ U t ^ and

V » V { N | N £ j , , N C . M } . By Lemma 5*7,

V - A {p(J) \ J z^ , J ^ V ) ; and by Lemma 5*6,

{J I J e , J V } « {J I J E , J ^ M } . Therefore,

Y « V {N I N £ Or , N ^ M }^M

" A{p(J) I J £ ^ , J ^ V }

« V*

Consequently, M E X •

A lattice is said to be meet-continuous provided that

for each M E and each net {M i a zA, M z iu)

irtiich is up-directed, then

^ J M A M ) « M A ( V M )* a £ ^ a a

A Join-continuous lattice is defined dually*

Consider the folloving properties of a lattice X* *

A*) «dls meet- (Join-) continuous*

B*) Every non-zero element of cu is a Join (nieet)

of completely Join- (meet-) irreducibles*

D*) dCis distributive.

Corollary 5*8; Let od e a complete lattice of closed

subspaces of E vith {o> and E contained injS . If. in ^

addition, jf satisfies properties A*), B.), and D*),

then X is T-reflexive for any topology T vhich is com­

patible vith the duality*

^oof: Properties A*) and D*) imply that tt is

infinitely meet- (Join-) distributive: Let M s / and

M £ X. , a £ 4 , and set M_ « V M^ vhere F is A ^ a e F «

a finite subset of sA * Since {Mj,} is up-directed, by meet-continuity and distributivity.

M A ( V Ma) « M A VM^ - V ( M A M - ) a F ^ F ^

« V [ V (M A M )] F a £ F

= V (M A M ).

Thus X Is infinitely meet-distributive*

Corollary 5*9: Let X ^e a complete, atomic Boolean

algebra of closed subspaces of E vith {0} , E E^d .

Then ^ is reflexive.

Proof: Since the atoms are compact, then od is a

complete algebraic lattice. Hence, ^dls meet-continuous.

Corollary 5*10: A finite distributive lattice of closed

subspaces is reflexive*

For conpleteness ve Include the folloving result

52 due to R* E* Johnson [10]:

Theoron; A finite lattice of closed subspaces of a

finite dimensional space is reflexive if and only if it

is distributive*

CONCLUSION

In Chapter II the lattice of edl reflexive sublattices

of subspaces of a finite dimensioned space vas seen to sat­

isfy both chain conditions* A natural question vhich curises

is vhat is the nature of the lattice of reflexive sublattices

of subspaces vhen no restriction is placed on the dimension

of the space. What role do maximal reflexives playt Is

every reflexive lattice contained in a maximal reflexive

lattice?

Also, ve might ask whether Theorem 2.9 is true for

infinite dimensions. Since Proposition 2*6 holds vithout

€iny restrictions on the dimension of the space, the theorem

is true in one direction. That is, if a reflexive lattice

^ is atomic and AlgiL has minimum condition on left ideals,

then Alg «d is semisimple vith minimum condition on left

ideals. The question now becomes; Kiowing that Alg oL/ is

semisimple with minimum condition on left ideals , what can

be said about ^ ? One might also eliminate the hypothesis

that A l g ^ has minimum condition on left ideals. Then,

knowing that Alg^i^ is semisimple, what can be said, if

anything, about i^ itselfT All sorts of problems might be

generated in this way: If certain restrictions are placed

53

on Alg X » vhat can be said about ; and, conversely,

if conditions are placed on it , vhat can be said about

A l g ^ *

In Chapters IV and V proving that a lattice in a

certain class of lattices vas reflexive involved producing

a set d ^^ operators and then demonstrating that Lat d,

vas actually equal X * In every case, the set d con­

sisted of operators of rank one. Conceivably, there could

exist a reflexive lattice X. tor vhich A l g ^ does not

contain any operators of rank one* The technique em­

ployed here gives no indication as hov to handle such

situations*

It appears that finding reflexive lattices remains

a very formidable problem, even in the finite dimensional

case. Certainly for infinite dimensions, it is clear

that the existing methods and techniques are not

completely adequate for solving the problem*

5Ji