Factorization of unbounded weakly compact operators

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Rev.Acad.Canar.Cienc . . , VIII (Núm. 1), 107-112 (1996)
FACTORIZATION OF UNBOUNDED W~AKLY COMPACT OPERATORS
T. Alvarez.
Opto. Matematicas. Universidad de Oviedo. Calvo Sotelo sin 33007 Oviedo. Spain.
Abstract. In this paper the factorísation result of Davis, Figiel, Johnson and
Pelczynski [DFJP] from bounded operators is generalise for unbounded operators .
Resumen. En esta nota generalizamos el resultado de factorizacion de Davis,
Figiel, Johnson y Pelczynski [DFJP] para operadores acotados debilmente compactos
actuando entre espacios de Banach al caso general de operadores lineales debilmente
compactos no necesariamente continuos y actuando entre espacios normados arbitrarios.
Keywords: weakly compact operator, reflexiva space, factorization of unbounded
operator.
1. INTRODUCTION
Let X and Y be normed spaces and T is a linear operator with domain a linear
subspace D(T) of X and range R(T) contained in Y. The class of all such operators T will
be denotad by L(X,Y).
We shall denote by XT the vector space D(T) with the norm 11 xll T = llxll + llTxll and
the operator GT ( or simply G ) denotes the canonical injection from XT into D(T).
Continuous operators T with D(T) = X will be referred to as bounded; TG is thus a
bounded operator. The adjoint of T is the operator defined by T' : D(T') e Y' -+ D(T)'
where D(T') = {y' e Y' : y' T is continuous on D(T) } and T'y'(x) = y'(Tx) for xe D(T) and
y'e D(T') (11; 11.2.2].
We shall write T 0 = JyT where Jy is the natural injection of Y into its
completion v- • and J denotes the canonical isometry of a given normed space into its
seconddual.
lf T is continuous, then T- is the closure of T regarded as an element of L(X- ,Y-).
In particular (TG)-is a bounded operator in L((XT)-,v-).
lf M is a linear subspace of X we write T/M for the restriction of T to M; by the
usual convention T/M = T/Míl D(T).
The operator T is called closable if T has an extension T whose graph is G(T). lf T =
T then T is said to be closed. A closed operator T is called a 0+·operator (0.-operator,
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Fredholm operator) if R(T) is closed and its null space is finite dimensional (if
R(T) is closed and of finite codimension,if Te 0+ íl 0-). We call Ta F +- operator
(partially continuous) if there exists a linear subspace M of X of finite codimension such
that (T/M)-1 exists and is continuous (T/M is continuous) [4],and T is an F- -operator
if T'e 0+) [7].
1 will be recalled (see,e.g [3), [8), [9]) sorne facts about bounded operators. Let T
be bounded and X and Y be Banach spaces. Then T is callad weakly compact·if TBx is
relatively cr(Y,Y')-compact ( Bx denotes the closed unit ball of X) and the following
properties are equivalent:
(1) T is weakly compact.
(2) T' is cr(Y' ,Y)-cr(X'.x'') continuous.
(3) T"X"CJY.
(4) T' is weakly compact.
(5) T factors through a reflexiva space.
In the general case of Te L(X,Y) we observe that the second adjoint of T presents as
an operator T" : D(T") e D(T)" ~ D(T')' and thus R(T")C IQY" where a denotes the
quotient map of Y" onto Y"/D(T')J.. and 1 is the canonical isometry mapping of QY" onto
D(T')'.
The operator Te L(X,Y) is said to be weakly compact if TBo(T) is relatively
cr(Y, D(T'))-compact (1 ). Sorne characterizations analogous to (1 )-(2) and (3) in the
general case of an arbitrary operator T are given in (1) where also it is investigated the
connection between weak compactness of T and that of T'.
In the present papar we will analyze the equivalence of properties (1) and (5) in
the general situation.
Throughout the remainder of the papar Tel(X,Y) where X and Y are normed spaces.
2.FACTORIZATION OF AN UNBOUNDED WEAKLY COMPACT OPERATOR
Glven an operator Te L(X, Y) we shall obtain a factorisation T 0 = jA through a
Banach space Zp (1sps oo ) where j is a tauberian injection and AGT is a bounded
operator. We factorise T 0 instead of T in order to retain the character of the [DFJP)
factorisation. The space Zp is constructed by setting W:= {T 0x : llxllT ~ 1, xe D(T) } and
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then proceeding in the usual way from the gauges of the set Un= 2nw + 2-nsy-.
We have the bounded operators
,...,
and (TG) = jB is the usual factorisation. Hence T 0 = jA where A= BJX.y.G-1 so
that AG is bounded . Since T 0 GBXT is contained in (TGBxi" the gauges of the sets
obtained from W = T 0 GBXT coincide with those used in the [DFJP] construction, based on
"" (TG) B(XTl· Hence the following theorem:
2.1 Tbeorem.([DFJP] factorization of T 0 ). Let TE L(X, Y) be given . Then
corresponding to each 1~~00 there is a Banach space Zp anda factorisation
A: D(T)-+ Zp,
in which jp is a bounded Tauberian injection, AG is bounded, and (TG)-= jp(AG)-.
Moreover if p>1 then the second adjoint of jp is injective. The map jp coincides with the
Tauberian injection in the [DFJP] factorisation of (TG)- corresponding to p.
Proof. ( see [2))
2.2. Propos!t!on. Let T be dense/y defined and partially cbntinuous. Then T' is
weakly compact if ánd only if T0 lactors through a reflexiva space.
Proof. Suppose that T' is weakly compact . By 2.1 Theorem it is enough to show that
(TG)-factors through a reflexiva space and for this it suffices, by the bounded case, to
preve that ( (TG)"' )' = (TG)' is weakly compact. But T' = (G-1 )'(TG)' [11; lemma 2.4)
and since T is partialiy continuous, so G-1 and hence (G-1 )' is continuous [5; corol.5].
Moreover (G-1 )' = (G')-1 ( as D(T) is dense ) and consequently R(G') is closed [10;
11.3.1). Therefore G'T' = IR(G')(TG)' and the hypothesis on T' now gives G'T' is weakly
compact [1 ;prop. 2.3) and by [12; 4.6.9) we conclude that (TG)' is weakly compact as
required.
Conversely, assume that T 0 factors through a reflexiva space. Then since the maps
ip in the [DFJP) factorisation of T 0 and (TG)-coincide by 2.1 Theorem it follows that
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(TG)' is weakly compact and an argument similar to the preceding implication
shows that T' = (G-1 )'(TG)' is weakly compact.
2.3. Theorem.Let T be dense/y defined and partial/y continuous. Consider the
following properties:
(i) T is weak/y compact.
(ii) T0 factors through a reflexive space.
Then (i) => (ii), and if Y is complete then the two properties (i) and (ii) are
equivalent.
Proof. (i) => (ii). lt follows readily from the above proposition and that if T is
weakly compact then so is T' (1; corol. 4.4].
(ii)=> (i). Suppose that Y is Banach. Then by (1; Th. 2.7] it is sufficient to show
that T' is a(D(T'),Y)-a(D(T)', D(T)") continuous ( as D(T)" = D(T") by the continuity
of T' ). But if T satisfies (ii) then by 2.1 Theorem and the bounded result (4) ~ (5) we
deduce that (TG)' is a(Y',Y)-a((Xr)'. (Xrt) continuous (*) and since (G-1 )' is
continuous we have (G-1 )' is a(D(G-1 )', (D(G-1 )')')- a(D(T)', D(T)") continuous
(**) [13; p. 234. ex.9].
Let (y'cx) be a net in D(T') which is a(D(T'),Y)-convergent to sorne point y'eY'.
Since Y is complete and T' is continuous is D(T') a(Y',Y)-closed [6; Th. 5.8] and so,
y' e D(T'). Since T' = (G-1 )'(TG)' (11; lemma 2.4] it follows from (*) that
a((Xr)'. (Xr)")-lim(TG)'y'cx = a((Xr)'. (Xr)")-lim i(TG)'y'a = (TG)'y' = i(TG)'y'
where i is the inclusion of D(G-1 )' in (Xr)' and since i' is surjective (10; IV.1.2] we
deduce that a(D(G-1 )', (D(G-1 )')')-lim(TG)'y'a = (TG)'y' and by (**) we conclude
that a (D(T)',D(T)")-lim (G-1 )(TG) 'y'cx = a (D(T)', D(T)")-limT'y'a = T'y' as
required.
The following examples show that the properties " T weakly compact " and " T
factors through a reflexiva space " are not equivalen! for an arbitrary operator T.
There is a weakly compact operator T acting between Banach spaces such that T has
not a factorization through a reflexive space.
Let X= Y = L1 (0,1] and define T by D(T) = {fe L1 [0,1] : f' exists almos!
everywhere and f' eL1[0,1]}. Then T is densely defined and D(T') ={O} [7; ex. 2.1].
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Hence T is not partially continuos [5; Th.3]. Now, let M be dense subspace of D(T)
consisting of the absolutely continuous functions. Then T/M is a surjective closed
Fredholm operator with one-dimensional null space (1 O; Vl.3.1] and so by [7; Th. 4.3]
(T/M)GT/M is a F_-operator. Hence ((T/M)GT/M)' is a F+-operator with non
reflexiva domain and by (1; prop. 3.5] is not weakly compact.
Suppose that T factors through a reflexiva space, then by 2.1 Theorem and the
bounded case it follows that ((T/M)GT/Ml' is weakly compact; a contradiction.
There exists an operator T which factors through a ref/exive space but is not
weak/y compact.
Let T be an unbounded closable operator in L(X,Y) with Y reflexiva. Then D(T') is a
dense proper subspace of Y' [6; Remark 5.12) and hence T is not partially continuous
since adjoints of partially continuos are continuous [5; Cor. 5). Moreover it is clear
that T factors through a reflexiva space .However T is not weakly compact since if Y is
complete and T is closable then T is weakly compact if and only if T' is weakly compact
( and then T is continuous ) (1 ; cor. 4.3). Let X= L 1(O,1 ), Y= L2[0, 1) and define T and
M as the previous example . Then T/M is surjective closed Fredholm operator with
one-dimensional null space (11; Vl.3.1 ). In particular T/M is an unbounded closable
operator.
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Recibido: 26 Diciembre 1996
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