I nternat. J. ath. & lath. Sci.
Vol. 5 No. 3 (1982) 529-536
529
A CHARACTERIZATION OF SINGULAR ENDOMORPHISMS
OF A BARRELLED PTAK SPACE
DAMIR FRANEKIC
1304 Spring Street
Bethlehem, PA 18018
U.S.A.
(Received October 29, 1980 and in revised form July 2, 1981)
ABSTRACT.
The concept of topological divisor of zero has been extended to endo-
morphisms of a locally convex topological vector space (LCTVS).
A characterization
of singular endomorphisms, similar to that of Yood [i], is obtained for endomor-
phisms of a barrelled
Ptk (fully complete)
space and it is shown that each such
endomorphism is a topological divisor of zero.
Furthermore, properties of the ad-
joint of an endomorphism are characterized in terms of topological divisors of zero,
and the effect of change of operator topology on such a characterization is given.
KEY WORDS AND PHRASES. Singul endmorphms, topological divisors of e,,
locally nvex barrelled Ptk spaces.
AMS (MOS) SUBJECT CLASSIFICATION (1980).
Primary 46H05, 46H20.
INTRODUCT ION.
1.
The reader should be familiar with barrelled spaces and have available the
four references listed.
IX, T1
and
The following notation and definitions will be used.
is an LCTVS over a field
C{X, TI
_
K
of complex numbers,
X’
is its topological dual,
the algebra of all the T-continuous endomorphisms of
the weak topology on
X by X’,
w*
oIX’,X),
and
8’
X.
oIX,X’)
w
is the topology on
X’
is
of uniform
convergence on all the w-bounded subsets of X--the strong topology.
CIX,TI
A
C[X,w)
can be made into a topological space in a number of ways.
is a family of w-bounded (hence bounded) subsets of
borhood base at zero, then the sets
X and N
NIT)
If
is a T-neigh-
530
D. FRANEKIC
{TgC{X, TI
N(A,V)
A
where A and V run through
and
N
respectively, form a neighborhood base at zero
IA, TI
for a locally convex vector topology
uniform convergence on members of
A
X
bounded subsets of
TA
The topology
i.e.
_
CIX’,TA)
_c
_c
if A
X,
X’
E, A
of B __c
the operator topology of
The space
C{X’,’).
F
C{X’,’).
X, {B,T)
T
If
B’.
A
then
is the operator topology of point-
while in the case of
A
B
the class of all w-
is the strong operator topology relative to
similarly B
is its absolute polar,
Tg(C(X,[),
DEFINITION i.
T.
Finally
is the absolute polar, in
o
(A,[))
and (D,-<) a directed set, the following definition
C(X,[).
T is a left (right) topological divisor of zero, itdz(rtdz), if
there is a net
eD} c__ C(X,T) which doesn’t converge
yet the net
does converge to zero, written
{$6:
TS6(S6T)
REMARK.
This means, there are
A
yet for all A e
and V e
N,
A’
g
A
and
TSA c__ (S6TA_V)
Following Yood [i] we use
Z(Z r)
V’
(r)
ze’ro,
written
TS6 O(ST+0),
such that
S
0,
in
SA’
V’ frequently,
to denote the sets of all left (right) tdz
their respective complements in
Finally,
g
to
eventually.
CIX, T).
the sets of all left (right) regular elements of
ments.
is then stronger
TgC{X,w), TA
If for each
extends the concept of topological divisors of zero (tdz) to
/IH r)
A
X.
X’.
Given
and
be a family of
of all the finite subsets of
F, {F,T)
A
A
Let
of uniform convergence on the members of
wise convergence relative to
bounded subsets of
T.
TgC{X’,w*)
than w*, in fact it is between w* and
C{X’,w*)
CIX,TI,
For each TgC(X,w) its adjoint is a w*-
which contains the family
X’
on
X’,
on
relative to
can be topologized in a similar manner.
continuous endomorphism on
V},
TA
Furthermore,
C(X,T)
and
IG r)
S(S r)
will mean
their comple-
are the sets of all left (right) divisors of zero
idz
(rdz).
2.
BAS IC RESULTS.
It can easily be seen that all the basic properties of tdz remain valid as in
the case of a Banach space.
LEMMA I.
Some of them are listed in the following lemma.
The following inclusions are valid in
C(X,[)
_
PTK SPACE
SINGULAR ENDOMORPHISMS OF BARRELLED
a)
Z l,
Nr
l,
S1
Gr
Gr
G
e)
H
Sr_N
n
d)
_
/
G/ Hr, Gr
c)
r
!Z
Z r Sr
m S
b)
531
Hr
Gr n
A slighz modification in Yood’s proof of Theorems 3.1 and 3.2 ([i], p. 493) yields
the follpwing result.
PROPOSITION i.
a)
r
b)
{T:
T is not injective}.
{:
+ X).
We shall refer to
TEC{X,[)
as to a topological isomorphism if T is injective
X
and I-relatively open as a p from
For yEX and
x’EX’
we define
TX.
onto
yx’EC(X,[)
yx’ (x)
by
x’(x)y.
The next theorem characterizes topological isomorphism in tes of tdz.
TEC(X,T}
THEOM i.
PROOF.
(S},
A’
and
T-topological isomorphism iff
is a
A I-topological isorphism can not be an itdz.
V’ as in Rerk after Definition i,
eventually for all
AeA
and
would, however, imply that
YEN,
TSGA’
S6A’
U
TV’
V’ frequently
TV’
particularly for U
For if it were with
yet
TSA
because T is open.
eventually which is impossible.
V
This
If T is
T-topological isomorphism, T is either not injeative, or T is injective but
-I
T
is not T-relatively open. T bein not injective implies, accordingly to Lena
-I
On the other hand, T
Z
la,
bein not I-relatively open implies the
not a
T Z.
.
existence of a net
and
x
0 in
Then the net
take
AEA
and
{S}
VEN,
TSA
TX
Let 0
+ x’EX’
C{X,[)
{x T-ly6}
with the property that
and construct the endomorphisms
is such that
TSA
then
and the net
x’ (A)yG.
suP{I<a,x’> :aEA}
imply that M
heDce
{y6}
S
0 yet
TS
in
y
S x2x’.
(A,[}.
To see this,
[-bodedness of A and T-continuity of
exists.
en
y
implies that
0
y
x’
M-Iv eventually,
V eventually.
COROLLARY i.
THEO 2.
T
TEC(X,w)
(C{X,w), {A,w))
is surjectlve iff
iff T is a w-topological isomorphism.
TEH r
{C{X,w), {,)).
532
A’
PROOF
Let T be surjective
VgF and
V’cN
_
FRANEKI
D.
r
If Tg2
SF
V’ frequently
STF 1 SF0 _c V’
net
S.
eventually for F
as in Remark, we have
VN,
ually for all EgF and
dicts the choice of the
(C(X,wl, T(F,T)),
X,
in
X,
{y
Let 0
{$6}
T’
its adjoint
c_q
T’’
#
x
X
_c
Z
STF c_ V eventT-IF, which contra-
Since
1
r.
If however the range of T is dense
is injective but not we-relatively open, hence there is a net
and the net
O
{x’
(T’)
S
y
such that
and construct the endomorphisms
has the property that
map (,\,x)
TENr
according to Lemma ib,
S,
In the case that its range is not
Conversely, assume T is not surjective.
dense in
then with
0 in
(F,T)
yet
C(X, II.
Xo@X 6
$6
ST
The net
The
0 in the same topology.
This proves that TgZ
Ax is separately continuous.
0
0 in w*-topology yet x
r.
The next four lemmas will be used to sharpen the results obtained so far.
A relatively open, continuous endomorphism of a barrelled space must
LEMMA 2.
have a barrelled range.
PROOF.
TX
If
{y6}
is not barrelled there is a net
and doesn’t belong to a barrel B in
TX.
_c
which tends to zero
T-lye}
{x
Then the net
zero because it is not eventually in the neighborhood U
LEM}iA
TX
T
-i
can not tend to
B.
Any relatively open endomorphism T of a complete space must have a
3.
closed range.
PROOF.
{y}
Let
be a net in the range of T and let
T-ly6}
Cauchy net and T is relatively open the net {x 6
converges to some xgX.
LEMMA 4.
adjoint
T’
PROOF.
X’.
Then
Tx6
Y6
Tx
Since
{y6}
is a
is also a Cauchy net hence
y, hence the range of T is closed.
Let T be an endomorphism of a barrelled
$’-topological isomorphism, then T
is
y.
y6
Ptk space (X,[).
If its
is surjective and open.
Let M be a balanced, convex, w*-closed and [-equicontinuous subset of
It is then both w*-compact and
is $’-bounded.
’-bounded.
B is also w*-closed because
T’
Since
T’
is
’-open,
is w*-continuous.
(T’)-IM
Since
(X,[I
B
is
barrelled, B is w*-compact and this in conjunction with w*-continuity of T’implies
that
T’B
T’X’ n
M is w*-compact, hence w*-closed.
space it implies that T is relatively open.
Since
{X,[)
is also a
Finally, completeness of a
Ptk
Ptk space
SINGULAR ENDOMORPHISMS OF BARRELLED
T’
and injectiveity of
imply that T
is both closed
PTK SPACE
533
X,
(Lemma 3) and dense in
hence
T is surjective.
LEMMA 5.
ly.
TCIX, T)
If
PROOF.
If Tg7
T-IA
A
1
r,
is surjective and
then with
$6,
eventually and this is impossible because
IX, Y)
If
TX
iff T is injective and
STA1
Ptk
TH
space, then
is barrelled.
_
SA’
in
(A,T),
TgH
r.
._
_
V’ frequent-
S6TA1
V’
(C(X,T), {AT))
A T-continuous injection from a Ptk space into a barrelled space is
PROOF.
a
is a barrelled
S6T 0
S6A’, hence
Since
IA,TI).
n {CIX,T),
A’gA and V’gN as in Remark,
is bounded, because T is open.
THEOREM 3.
THr
Y-open, then
T-topological isomorphism, hence according
to Theorem i,
T
follows from Lemma 2 and Theorem i.
THEOREM 4.
{XT)
If
is a barrelled
Ptk space,
Tr
then
The converse
(C(,T), (A,T))
iff T is surjective.
PROOF.
T-open
If T is surjective then, according to Proposition 2
[3, p. 299], T
is
hence by Lemma 5 it cannot be a rtdz.
Asume
now that T is not surjective.
according to Lemma 1 a,
T-open,
can not be
Tg r
r.
If the range of T is not dense, then
Suppose that the range of T is dense in
because it would have to have a closed range (Lemma 3).
ing to Lemma 4, its adjoint
hence there is a net
{y}
T’ (which
T’X’ ,Y
X.
T
Accord-
is injective) can not be $’-relatively open,
0 and the net
(T’)-ly}
{x
0 in
6’
The conclusion then follows just as in the last part of Theorem 2.
Since Fr$chet space is both barrelled and
valid for them.
COROLLARY 2.
If
{X,I)
is a
_
Frchet space
Ptk
then
a)
T is injective and range closed iff
b)
T is surjective iff
PROOF.
Tgr
space, Theorems 3 and 4 are
Tg
(C{X,T)
(A,T)
(C(X,I), (A,II).
Part a) follows from the fact that a closed subspace of an Frchet
space is a Fr$chet space, hence barrelled and from Theorem 3.
Part b) follows
from Theorem 4 and the foregoing remark.
Properties of a linear operator on an LCS are very intimately related to those
534
FRANEKI
D.
"T
For example:
of its adjoint operator.
is w-open iff
T’
The next theorem relates properties of T with those of
T’
in terms of tdz.
In this regard it is important to note that an operator topology
A
unaltered if either
closed".
is w*-range
{A,TI
remains
is replaced with its balanced convex and closed envelope
and/or N is replaced with a fundamental system of balanced, convex and closed
In the sequel it is assumed that each
neighborhoods.
vex and closed.
THEOREM 5.
Let
A
__
AsA
and
VsN
X
and
TsC{X,w}
be a family of bounded subsets of
is balanced con-
such that
Then the following are equivalent:
TZI(Z r)
T’glr{l/}
a)
b)
(C(X,w}, %(A,T)).
continuous subsets of
PROOF.
The condition
TA-boundedness
EE
of each
T(E,TA))
(C(X’,w*),
TA
_c
A
where
E
is the set of all
T equi-
X’.
makes
implies that
T’
TA-continuous.
T(E,T A)
This together with the
is a vector topology.
The state-
ment then follows from the following facts:
TA _c B iff
S
0 in
T’B
A
(A,T)
iff
if
S
are
convex, balanced and closed;
{A,TA)7
0 in
T’S
(ST)’
In what will follow B’ and
subsets of
A,B
F’
will denote the sets of all w*-bounded and finite
X’ respectively.
There is a number of fanilies of bounded subsets (equivalently w-bounded) of
which satisfy the condition of Theorem 5 for all
F--the family of
extreme ones:
that
F
c__
F
all finite--and
generates the w*-topology and
B, T{F,w)
<
T{B,wl.
B
the
TgC{X,w).
Consider the most
B--all bounded--subsets of X.
B’-topology
in
X’.
In
CIX, wl,
a)
b)
c)
If
Tzl(z r)
TZI(Z r)
Tzl(z r)
Note
since
The effect of change of operator topology on the results
__
of the preceding theorem are given in the following corollary
COROLLARY 3.
X
_
TgC(X, wl, the following
statements are valid:
(C(X,w), (F,w))
iff
Tzr(zl)
(C(X,w), ?(B,w))
iff
Tzr(z l)
(C(X,w), %(B,T))
iff
Tkzr(z l)
c__
(C(X’
W
(F’,w*)
(C(X,w*),
(C(X’,w*), T(E,B’)).
SINGULAR ENDOMORPHISMS OF BARRELLED
PTK SPACE
535
The next theorem characterizes w-isomorphisms and their adjoint in terms of tdz.
THEOREM 6.
TeC{X,wl,
If
the following are equivalent:
a)
T is a w-topological isomorphism.
b)
T’
c)
TeH1
d)
T’gH r
PROOF.
is surjective.
Z(B,wl)
(C(X,w)
(C(X’ w*)
%(F’
The equivalence of a) and b) is a standard result and could be found
for example in [2, Proposition 8.6.3, p. 517].
The equivalence of a) and c) follows
from Corollary i while that one of c) and d) from Corollary 3b.
Frchet space
The preceeding results can be strengthened in the case of
"T
the fact that:
is an isomorphism iff it is a w-isomorphism"
due to
[2, Theorem 8.6.13,
p. 521 ].
COROLLARY 4.
Frchet space, the following
(C(X,T), %(B,w) ).
b)
TgZ/=
(C(X,T), "7(B,T) ).
c)
T’g Z r
(C(X’,w*), (F’,6’)).
r
(C(X’ ,w*), ?’(B ,6 )).
PROOF.
T’Z
F’.
are equivalent:
The equivalence of a) and b) is by Theorem 6, the foregoing remark
c) is equivalent to d) because in the dual of a barrelled space
and Theorem i.
B’
is a
a)
d)
E
(X,T)
If
TEZ1
Finally, b) is equivalent to d) by Corollary 3c.
The next theorem generalizes the following result of Rickart
"A singular endomorphism
THEOREM 7.
[4, p. 297]
of a Banach space is a topological divisor of
zero."
Every singular endomorphism of a barrelled Ptk space is a topolo-
gical divisor of zero.
PROOF
T is singular iff T is either not injective in which case
or T is not surjective.
TgNr
3
c__
Z
r.
Tg
zl
In the latter case, if the range of T is not dense,
If however, the range of T is dense then according to Theorem 4, TgZ
r.
CONCLUS ION.
Preliminary results, obtained in this paper, indicate that the concept of tdz
can successfully be used to classify endomorphisms of LCS which are of a more gen-
D.
536
FRANEKI
eral nature than is Banach space.
A decomposition of
CIX,T)
[i], has not been attempted.
into nine disjoint subsets such as in Theorem 3.14,
The conjecture is that it is possible.
A difficult question seems to be the one in regard to a topological characterization of the set of regular endomorphisms and others.
An answer to it seems to
be directly related to the question:
Under which conditions is
C(X, TI
a topological algebra with a continuous in-
verse?
ACKNOWLEDGEMENT.
This is an extended version of a portion of the author’s disser-
tation written under Professor A. Wilansky at Lehigh University.
REFERENCES
I.
YOOD, B.
2.
EDWARDS, R.E.
3.
HORVATH, J.
Transformations between Banach space in the uniform topology, Annals
of Mathematics, Vol. 50, No. 2, April 1949.
Functional Analysis.
Holt, Rinehart and Winston, 1965.
Topological Vector Spaces and Distributions, Vol. I.
Addison-
Wesley Publishing Company.
4.
RICKART, C.E.
General Theory of Banach Algebras.
D. Van-Nostrand Co.