^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