Strictly singular and strictly cosingular operators on spaces of continuous functions

Abstract

In this paper we will be concerned with studying operators T: C(K, X) → Y defined on Banach spaces of continuous functions. We will be particularly interested in studying the classes of strictly singular and strictly cosingular operators. In the process, we obtain answers to certain questions recently raised by Bombal and Porras in [5]. Specifically, we study Banach space X and Y for which an operator T: C(K, X) → Y with representing measure m is strictly singular (strictly cosingular) whenever m is strongly bounded and m(A) is strictly singular (strictly cosingular) for each Borel subset A of K. Along the way we establish several results dealing with non-compact operators on continuous function spaces, and we consolidate numerous results concerning extension theorems for operators defined on these same spaces. Also, we join Saab and Saab [25] in demonstrating that if l1 does not embed in X* then the adjoint T* of a strongly bounded map must be weakly precompact, thereby presenting an alternative solution to a question raised in [2].

Some Classes of Continuous Operators on Spaces of Bounded Vector-Valued Continuous Functions with the Strict Topology

“…Bilyeu and Lewis [21,Theorem 4.1] showed that if is compact, then every strictly singular operator : ( , ) → is strongly bounded and, for each Borel set in , ( ) : → is strictly singular. Strictly singular operators : ( , ) → have been studied by Bessaga and Pełczyński [44] and Abbott et al [18]. Now we show an analogue of Theorem 4.1 of [21] for ( , ‖ ⋅ ‖)-continuous and strictly singular operators : ( , ) → , where is a completely regular Hausdorff space.…”

“…( , ) → (resp., : 0 ( , ) → ) have been studied intensively; see [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. The study of the relationship between operators : ( , ) → (resp., : 0 ( , ) → ) and their representing operatorvalued measures is a central problem in the theory.…”

Strongly Bounded Representing Measures and Convergence Theorems

“…If P : X → 1 is a projection, then P is not compact. By Theorem 2.2 of [2], there is a compact space and a continuous linear surjection m ↔ T : C( , X) → 1 so that m is strongly bounded and m(A) : X → Y is compact for each A ∈ . Since T is a surjection onto 1 …”

On Convergence of Conditional Expectation Operators

“…Given such an operator T , a finite partition π of H, and a measure µ on H, a conditional expectation operator T π can be defined. It was shown in [1]that the net (T π ) directed by refinement does not always converge to T in the operator norm.…”