Operators Having Weakly Precompact Adjoints

Abstract

Throughout this note, X , Y, E and F will denote real BANACH spaces. The unit ball (sphere) of the BANACH space X will be denoted by B,(S,), and the term operator will mean a bounded linear function. An operator T: X + Y is said to be weakly precompact (wpc) if T(B,) is weakly precompact, i.e., (T(x,)) has a weakly CAUCHY subsequence for each sequence (x,) from B,. It follows easily from ROSENTHAL' S fundamental ['-theorem [R] that an operator T: X + Y is wpc if and only if TI, is not an isomorphism for any subspace Z of X isomorphic to e'. In this paper we will be concerned with studying operators T: X + Ysuch that the continuous adjoints T* : Y* + X * are wpc. The specific motivation for our results come from BOMBAL and PORRAS [BPI, ABBOTT, BATOR, BILYEU, and LEWIS [ABBL], ABBOTT, BATOR, and LEWIS [ABL], and BOMBAL and CEMBRANOS [BC].Before stating our first result, we establish some additional notation and terminology. If T : X + Yis an operator, then a sequence (y:) in Y* is said to be T-weak*-null if (T(x), y,*) -+0 for each x EX. The sequence (en) will denote the canonical unit vector basis of c, , , and (e:) will denote the unit vector basis of 4'. If each of (x,) and (y,) is a basic sequence, we write (x,) -(y,) to signify that they are equivalent, and we denote the closed linear span of a subset A of X by [A]. The reader may consult DIESTEL [D] or [ABL] for any unexplained notation.Our first result establishes a connection between wpc adjoints and strict singularity.Theorem 1. If T: X + Y is an operator, T* : X * -+ Y* is wpc, and U is a BANACH subspace of X so that the restriction of T to U is an isomorphism, then U* does not contain t?'.

Strongly Bounded Representing Measures and Convergence Theorems

“…SinceT is an extension of T, there is a subspace W of B( , X) so that ST(W ) = c 0 . Thus by Theorem 4 of [6],T * is not weakly precompact, and we have a contradiction.…”

“…By Corollary 6 of [6], T * x is weakly precompact. Hence T x is an unconditionally converging operator on a C(K)-space, and every unconditionally converging operator on a C(K)-space is weakly compact [14,27] Proof.…”

“…Hence T x is an unconditionally converging operator on a C(K)-space, and every unconditionally converging operator on a C(K)-space is weakly compact [14,27] Proof. IfT * is weakly precompact, thenT is unconditionally converging and weakly precompact [6]. Hence T is strongly bounded andT : B( , X) → Y .…”

“…Corollary 2 of [6] shows that if T * : Y * → X * is weakly precompact, then T : X → Y is unconditionally converging and weakly precompact. It follows that if T : C(K, X) → Y has a weakly precompact adjoint, then T is strongly bounded (since it is unconditionally converging) and T * is unconditionally converging (by Theorem 12).…”

“…Hence T is strongly bounded andT : B( , X) → Y . Apply Theorem 4 of [6] to obtain a subspace Z of C(K, X) and an operator S : Y → ∞ so that ST(Z) = c 0 . SinceT is an extension of T, there is a subspace W of B( , X) so that ST(W ) = c 0 .…”