Rev.Acad.Canar.Cienc., I, 109-117 (1990)
ON THE GENERALIZED SADOVSKll FUNCTOR 1
Manuel Gonzalez
Departamento de Matem~ticas, Universidad de Cantabria
39005 Santander (Spain)
and
Antonio Martln6n
Departamento de Analisis Matematico, Universidad de La Laguna
38071 La Laguna, Tenerife (Spain)
ABSTRACT
For an ideal of operators 1', we introduce a functor PA which associates
to a Banach space X another Banach space P ,ix> and to a (continuous linear)
operator T another operator P
14(T). Particular cases of the functor P i4 have
been considered by several authors. By using a generalized measure of
noncompactness we obtain an expression of the norm of an element of P ,iX).
Moreover characterize the tauberian operators as those operators for P WCo (T)
is one-one, WCo the ideal of weakly compact operators.
CLASSIFICATION AMS (1980): 47030
KEY WORDS: operator ideal, measure of noncompactness, tauberian operator
1 Supported in part by DGICYT grant PB88-0417.
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INTRODUCTION
Independently, F. Quigley (see (RI, pp. 25- 27)) and S. Berberian (BE)
esentially considered for a Banach space X the quotient of the space of all
bounded sequences in X, l
00
(X), by the subspace of all sequences converging to
Q(X) := l (X)/c (X) .
(X) 0
The norm of a coset (x )+c (X) in Q(X) has the expresion (HAiJ
n o
If T: X ----) Y is an operator, then T is induces another operator
Q(T): Q(X) ----) Q(Y) .
R.E. Harte [HAl) calls Q the Berberian-Quigley functor. For applications see
[BE), (BHW), [HAI), (HA2), (RI].
B.N. Sadovskii [SA), unawer of the works of F. Quigley and S. Berberian,
defines the functor P taking the subspace of l
00
(X) of all sequences with
relatively compact range m(X). Then
P(X) := l (X)/m(X) .
(I)
The norm of a coset (xn)+m(X) in P(X) has the expression (HW)
ll(xn)+m(X)ll = h({xn}) ,
where h({xn}) is the Hausdorff measure of noncompactness (BG) of {xn}' the
range of (xn) . If T: X ----) Y is an operator, then T induces another operator
P(T): P(X) ----) P(Y) .
We call P the Sadovskii functor . Independently, it has been considered by
J. J. Buoni, R. Harte and T. Wickstead (BHW). For applications see [AKPS),
[BHW), (CW), [FA), [HW), (SA), [TY), {ZEI.
J.J . Buoni and A. Klein [BK) define the functor Pw taking the subspace
of l (X) of all sequences of relatively weakly compact range, m w(X). Then
(I)
Analogously to Q and P, if T: X ~ Y is an operator, then we can define
another operator
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In this paper we show that the norm of a coset (x )+m w(X) in Pw(X) has
n
the expression
where w is the measure weak of noncompactness, defined by F. S. De Blasi (DB].
We obtain this result as a particular case of a general construction in
which, for every operator ideal A, we introduce a generalized Sadovskii
functor. We study this functor by means of a set measure introduced by K.
Astala. We also obtain a characterization of tauberian operators by means of
the functor associated to the weakly compact operators.
THE AST ALA MEASURE
Recall that a class A of (linear and continuous) operators between
Banach spaces (over the real or the complex field 110 is an ideal of operators
if the following conditions hold (PI):
(1) The identity operator of IK, IIK, belongs to A.
(2) The class A(X, Y) of all operators in A between the Banach
spaces X and Y is a subspace of !e(X, Y), the class of all
operators between X and Y.
(3) If there exists RST, where R and T are operators, and S is an
operator in A, then RST belongs to A.
An ideal A is called closed if each component A(X, Y) is closed in
!e(X, Y) . A is called sur jective if for any sur jective operator Qe!e(Z,X), an
operator TE!e(X, Y) belongs to A whenever TQEA(Z, Y) . We use the notation ,a~ for
the smallest sur jective closed ideal containing A.
In the following A will be an ideal of operators; X, Y, Z Banach spaces;
IB the c lass of all Banach spaces; BX the closed unit ball of XEIB.
By using an ideal of operators A, K. Astala has given the following
def inition:
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l DEFINITION (AS; 3.1)
Let DcX bounded. Then h ,iD) is defined by
h A(D) := lnf {c>O : 3Zel8, 3Ke.il1(Z,X), DcKB
2
+cBX}
The set D is called A-compact if h A(D)=O.
We list in the following proposition the properties of h A that we use in
this paper.
i PROPOSITION (AS; 3.3 (b), (c); 3.5; 3.11)
Assume C, D are bounded subsets of X and AEIK is a scalar. Then:
(1) hA(C+D) :S h A(C)+h A(D)
(2) h,,f(AD) = IAl h,,f(D)
(3) h.4 = h.4"
(4)h A(D)=O <=> 3Ze18, 3KeA"(Z,X), DcKBZ
THE GENERALIZED SADOVSKII FUNCTOR
For a Banach space X we consider the space of all bounded sequences (xn)
in X:
attached with the norm
We consider the subclass of the sequences (xn) with range {xn} .A-
compact:
]. PROPOSITION
:= {(x )et (X)
n ro
mA(X) is a closed subspace of t
00
(X).
{x
n
nelN}
PROOF. By using Proposition 2 (1), (2), we obtain that mA(X) is a subspace of
l (X).
00
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Now we show that m A(X) is closed. From the definition it follows clearly
that h .. i<xn})~ll(xn)ll. Then h,a= l
00
(X) ---) R is continuous, where R is the real
- 1
field; hence m A(X)=h A (0) is closed. •
The following proposition assures that any operator T: X ---) Y maps
sequences of mA(X) into mA(Y).
i PROPOSITION
If Te.f(X, Y), then
{xn)emA(X) ~ (Txn)emA(Y) .
PROOF. As (xn)em A(X), there exist Zel8 and KeAA(Z,X) such that
{xn} c KB
2
(Proposition 2 (4)), and consequently (Txn}cTKB
2
. Since TKeAA(Z, Y), we
•
Now we can define a generalized Sadovskii functor. which also generalize
the Buoni-Klein functor (see Introduction).
.§. DEFINITION
We define the generalized Sadovskii functor PA associated to an ideal of
operators .4 in the following way
{l) XeB---) P A(Xl := l
00
(X)lm
111
CX) •
(2) Te.f(X. Yl ---) P A(T)((xn)+m A(X)) := (Txnl+m A(Y) •
where
By Proposition 3, P .sd(X) is a Banach space. Moreover,
Te!i'.(X, Yl ~ P A(T)e!l'.(P A(X),P A(Y)) .
In fact, let (x lei (X) be such that ll(x )11~1. Then n oo n
llP A(T)((xnl+m.,iX))ll = ll(Txn)+mA(Y)fl :S ll(Txn)ll
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= sup { llTxn II : nEIN} ::s llTll .
Hence P A(T)ef(P A(X),P A(Y)) and llP A(T)ll ::s llTll.
By using the definition of PA we obtain the following proposition.
§. PROPOSITION
Assume T,Se!e(X,Y), Re!e(Y,Z) and ;\.ellC. Then
(1) P A(T+S) = P A(T)+P A(S)
(2) P A(H) = ;\. P A(T)
(3) P A(RT) = P A(R)P A(T)
Now we give an expression for the norm of a coset (xn)+mA(X) in P A(X) by
using the Astala measure h .,a·
1 PROPOSITION
If (x )et (X), then n oo
hA({xn}) = ll(x
0
)+m,a(X)ll
PROOF. If c>hA({x
0
}), then there exist ZeE and KeA(Z,X) such that
{x
0
} c KB
2
+£BX ,
hence for every nE~ ther e are zne8
2
and bneBX verifying
and consequently llx
0
- Kz
0
11::sc, being (Kzn)em,iX) . Then ll(x~)+mA(X)ll::sc, hence
ll (x
0
)+mA(X)ll ::s hA({xn}) .
If c >ll( x
0
)+ mg/Xlll, there exists (y
0
)emA(X) such that
ll (x
0
-yn)ll < c ,
hence f or every nE~. x
0
- y n Ec8X, and consequently
{xn } c {y
0
}+ cBX c KB
2
+cBX ,
being ZEIB, KEllnZ,Xl, (yn k KB
2
(Proposition 2 (4)) . Then hA({xn})::sc, hence
h..d({ x
0
} )::sll (x
0
)+ mA(X ) ll . •
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!! REMARK
If A=Co, the compact operators, we obtain that hCo is the Hausdorff
measure of noncompactness (BG); mCo ls the class of all bounded sequences for
which every subsequence has a convergent subsequence; P Co ls the Sadovskii
functor (SA].
P Co (T) is one-one if and only if T is upper semi-Fredholm operator
(finitedimensional kernel and closed range). Moreover, P Co(T)=O if and only
if TeCo. The Proposition 7 for A=Co appears in (HW].
2. REMARK
If s4.=WCo, the weakly compact operators, we obtain what hWCo is the
measure of weak noncompactness, defined by De Blasi (DB); mWCo is the class
of all bounded sequences for which every subsequence has a weakly convergent
subsequence. The expression
hWCo({xn}) = ll(xn)+mWCo(X)ll
has not appeared in the literature.
P WCo is the Buoni-Klein functor (BK]. Our proof of Proposition 3 is more
simple than the proof of (BK] for mWCo'
Moreover (BK; Theorem 6) affirms the following:
Kernel of T reflexive and complemented, and range of T closed *
* P WCo (T) one-one ~
~ T have reflexive kernel
This result is weaker than Proposition 11 below. Also P WCo (T)=O if and only
if TeWCo.
C 11 AR A< I I:. K I /A I I 0 N 0 F T AU HI:. R I AN U I' 1-. RA ·1 0 R S
Let T*el(Y*,X*) be the conjugate operator of Tel(X, Y) and J(X) the
canonical image of X in the second dual xu. An operator Tel(X, Y) is said to
be tauberlan if its second conjugate Tu verifies
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T**-l J(Y) = J(X) .
Tauberian operators were introduced by N. Kalton and A. Wilansky [KW). We use
the following characterization of tauberian operators (GO]: Tef(X, Y) ls
tauberian if and only if given a bounded sequence (xn)' (Txn) weakly
convergent implies (xn) has a weakly convergent subsequence. We give a new
characterization of tauberian operators by using the functor P WCo.
12 PROPOSITION
If Tef(X, Y), then
T tauberian ~ P WCo (T) one-one
PROOF. If T is tauberian and
PWCo(T)((xn)+mWCo(X)) = mWCo(Y) '
then T(xn)emWCo(Y). Therefore, any subsequence of (xn) has a subsequence
weakly convergent. Hence (xn)emWCo(X). Conversely, if (Txn) ls weakly
convergent, then T(xn)emWCo(Y) and consequently (xn~EmWCo(X). Therefore there
exists a subsequence of (xn) which is weakly convergent and then T is
tauberian. •
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