Internat. J. Math. & Math. Sci.
VOL. 13 NO. 2 (1990) 271-274
271
A NOTE ON TAUBERIAN OPERATORS
JESS
ARAUJO
Departamento de Matematlcas
ETSI Industriale s
Unlversidad de Oviedo
Castiello de Bernueces
33204 GiJn, Spain
J. MARTINEZ-MAURICA
Departamento de Matematlcas
Facultad de Ciencias
Unlversidad de Santander
Av. Los Castro s
39071 Santander, Spain
(Received May 20, 1988 and in revised form August 23, 1988)
ABSTRACT.
In this note we prove the existence of operators which are not Tauberlan
even though they satisfy properties about restrictions being Tauberian.
The operators
are
non-reflexlve
defined
subspace.
on
Banach
spaces
which
contain
a
somewhat
reflexive,
This gives an answer to a question proposed by R. Neidinger [i].
KEY WORDS AND PHARSES. Tauberian Operators, and Semi-Fredholm Operators.
1980 AMS SUBJECT CLASSIFICATION CODES.
47A56.
I. INTRODUCTION.
Throughout this note E, F are infinite-dimensional Banach spaces over the real or
complex field.
Given T
All operators T:E
L(E,F) the notation
TIZ
F are assumed to be linear and continuous.
denotes the restriction of T to the subspace Z of E.
Recall that an operator T E L(E,F) is said to be semi-Fredholm if its null space
Also, a Tauberian
N(T), is finite-dimenslonal and its range space R(T) is closed.
a bounded linear
A.
is
D.
Wllansky in [2],
Garllng and
operator, as defined by
operator T E L(E,F) such that T" preserves the natural embedding of E into its double
E. Some relationships between these tw classes of
F implies x"
dual, i.e., T"x"
In particular, if R(T) is
operators have been studied in [i], [3], [4] and [5].
closed, then T is Tauberian if only if N(T) is reflexive.
It is well-known that the restriction of a semi-Fredholm operator to any closed
subspace
is
again
a
semi-Fredholm
operator.
In
the
opposite
direction
it
is
J. ARAUJO AND J. MARTINEZ-MAURICA
272
x)rthwhile to mention the
THEOREM I
that T:E----+ F
of
E
contains
Fredholm.
following result that is basically due to T. Kato [6],
[6]).
(c.f.
Let E,F be inflnlte-dlmenslonal Banach spaces.
Assume
is an operator such that every inflnlte-dlmenslonal closed subspace Z
an
inflntle-dlmenslonal
closed
subspace W for which
TIW
is
semi-
Then T is semi-Fredholm.
It follows that in order to see that a given operator T is semi-Fredholm, it is
enough to assure that its restriction to every closed subspace with a Schauder basis
is semi-Fr edholm.
Another related result is the following theorem due to R. Neldlnger in which
Banach spaces with no inflnlte-dlmenslonal reflexive subspace are called "purely non-
reflexive" spaces.
THEOREM 2 ([I], p. 26).
Let E be a weakly sequentially complete Banach space and
let T
L(E,F). Then T is Tauberlan if (and only if)
purely non-reflexlve closed subspaces Z of E.
TIZ
is semi-Fredholm for all
In view of the preceding theorem, R.
Neldlnger raised the following question
([I], p. 139): If T E L(E,F), restricted to any purely non-reflexlve closed subspace
is semi-Fredholm, is T Tauberlan?.
Indeed, the answer is positive if E is
we
reflexive.
that
E
assne
is
not
reflexive.
In this case there are some
Then,
trivial
situations
reflexive
space,
for which
that
Is,
the
every
answer is negative
inflnlte-dlmenslonal
(e.g., let E be a somewhat
subspace of E contains an
Inflnlte-dlmenslonal reflexive subspace, and let T be a finite rank operator).
Our
next example gives a negative answer to the question raised by R. Neldlnger in a non-
trivial situation.
EXAFPLE.
by T(x,y)
Let J be the James space and let T:J x
ii----+
I
I
be the operator defined
y.
J is not reflexive then, T is not Tauberlan.
Since R(T) is closed and N(T)
Now, let Z be a purely non-reflexlve closed subspace of J x i
Since J is somewhat
N(T) O Z is flnlte-dlmenslonal; otherrlse, N(TIZ would contain an
reflexive, N(TIZ)
I.
inflnlte-dlmenslonal reflexive subspace, which contradicts our assumption over Z.
Also, N(T) and Z are totally incomparable Banach spaces (i.e., there exists no
inflnlte-dlmenslonal Banach space which is isomorphic to a subspace of N(T) and to a
i
subspace of Z). This implies that N(T) + Z is closed in J x 1 [7] and hence,
R(TIZ is closed by the open mapping theorem.
Thus, TIZ is semi-Fredholm for all purely non-reflexive
T(Z)
closed subspaces.
2. MAIN RESULTS.
Another
related
problem
is
as
follows;
we
know
that
Tauberlan operator to any closed subspace is again Tauberlan.
for Tauberlan operators instead of semi-Fredholm operators?.
the
restriction
of
a
So, is Theorem 1 true
The answer is obvlously
positive if, for instance E is reflexlve or E is purely non-refelxlve.
However, we
have,
THEOREM 3.
Let E be an Inflnlte-dlmenslonal Banach space which contains an
Inflnlte-dlmenslonal somewhat reflexlve closed subspace M which is not reflexive.
TAUBERIAN OPERATORS
there
Then
an
exists
surjectlve operator T:E
273
Banach
inflnlte-dlmenslonal
space
F
and
of E contains an Inflnlte-dlmenslonal closed subspace W for which
PROOF.
First assume that
map T:E----+
E.
E/M
is Tauberlan.
It follows, as in the above example, that T is not Tauberlan but
that Z is not purely non-reflexlve;
TIZ
Now,
is Tauberlan.
in this case there exists an infinite-
For this W,
Z.
subspace W
reflexive
dimensional
TIW
is inflnlte-dlmenslonal and consider the quotient
that for every purely non-reflexlve subspace Z of E then,
assume
non-Tauberlan
a
F such that every inflntle-dlmenslonal closed subspace Z
it
is
obvious
that
is
Tauberlan.
If dim E/M
<
is not reflexive, there exists a bounded basic sequence
null [8].
(e2n_l).
(e2n)
Without loss of generality,
(en)
in E which is not weakly
is not weakly null, otherwise use
(e2n).
Let N be the closed linear span of
It follows that N is a non-
reflexive closed subspace of E such that E/N is inflnlte-dlmenslonal.
that
Since E
then, E is itself somewhat reflexive and non-reflexlve.
the quotient map T:E -----+ E/N satisfies
the
conclusion.
Let us prove
Given
an
infinite-
dimensional closed subspace Z of E then, Z contains an inflnlte-dlmenslonal reflexive
subspace W;
it
follows that T
IW
is
Tauberlan.
But, on the other hand, T is not
Tauberlan because its null space N is not reflexive.
ACKNOWLEDGEMENT.
We are grateful to the referee for some valuable suggestions.
REFERENCES
I.
NEIDINGER
R.,
Properties
Tauberlan
of
Operators
on
Banaeh
spaces,
Ph.
D.
Dissertation, University of Texas at Austin, 1984.
2.
GARLING
3.
KALTON N.
D.J.}I.
and WILANSKY A.,
On a summabillty theorem of Berg, Crawford and
Whltley, Proc. Cambridge Philos. Soc. 71(1972), 492-497.
and WILANSKY A.,
Tauberlan operators on Banach spaces,
Proc. Amer.
Math Soc. 57(1976), 251-255.
4.
NEIDINGER
R.
and
ROSENTHAL
H.P.
Norm-attalnment
of
linear
functlonals
on
subspaces and characterizations of Tauberlan operators, Pacific J. of Math.
5.
6.
7.
8.
118(1985) 215-228.
Soc.
Math.
Trans. e r.
YANG
K.M., The generalized Fr edholm operators,
216(197 6), 313-326.
KATO T., Perturbation theory for nullity deficiency and other quantities of
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ROSENTHAL H.P., On totally incomparable Banach spaces, J. Funct. Anal. 4(1969)
167-175.
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