Compactness and sequential completeness in some spaces of operators

Marian Nowak

doi:10.1007/s11117-013-0248-2

Abstract

Let $$X$$ be a completely regular Hausdorff space and $$C_b(X)$$ be the Banach lattice of all real-valued bounded continuous functions on $$X$$ , endowed with the strict topologies $$\beta _\sigma ,$$ $$\beta _\tau $$ and $$\beta _t$$ . Let $$\mathcal{L}_{\beta _z,\xi }(C_b(X),E)$$ $$(z=\sigma ,\tau ,t)$$ stand for the space of all $$(\beta _z,\xi )$$ -continuous linear operators from $$C_b(X)$$ to a locally convex Hausdorff space $$(E,\xi ),$$ provided with the topology $$\mathcal{T}_s$$ of simple convergence. We characterize relative $$\mathcal{T}_s$$ -compactness in $$\mathcal{L}_{\beta _z,\xi }(C_b(X),E)$$ in terms of the representing Baire vector measures. It is shown that if $$(E,\xi )$$ is sequentially complete, then the spaces $$(\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)$$ are sequentially complete whenever $$z=\sigma $$ ; $$z=\tau $$ and $$X$$ is paracompact; $$z=t$$ and $$X$$ is paracompact and Čech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on $$C_b(X)$$ is given.

Highlights

Introduction and terminologyFor terminology concerning vector lattices we refer the reader to [1]
We denote by σ (L, K ), τ (L, K ) and β(L, K ) the weak topology, the Mackey topology and the strong topology on L, with respect to a dual pair L, K
On we assume that X is a completely regular Hausdorff space

Summary

Introduction and terminology

For terminology concerning vector lattices we refer the reader to [1]. We denote by σ (L , K ), τ (L , K ) and β(L , K ) the weak topology, the Mackey topology and the strong topology on L, with respect to a dual pair L , K. Graves and Ruess [6, Theorem 7] characterized relative compactness in ca( , E) (= the space of all E-valued countably additive measures on a σ -algebra ) in the topology Ts of simple convergence (convergence on each A ∈ ) in terms of the properties of the integration operators from S( ) to E and from L( ) to E. [14, Theorem 3.4]) we study relative Ts-compactness in the space Lτ,ξ (B( ), E) of all (τ (B( ), ca( )), ξ )-continuous linear operators from B( ) to E Let ξε stand for the topology on Eξ of uniform convergence on all sets A ∈ Eξ , i.e., ξε is generated by the family of seminorms {qA : A ∈ Eξ }, where qA(e ) = sup{|e (e )| : e ∈ A} for e ∈ Eξ ,. For each e ∈ Eξ let (m T )e ( A) := m T ( A)(e ) for all A ∈ B

From the general properties of the operator Tit follows immediately that

Note that

It follows that supT