# Compactness and sequential completeness in some spaces of operators

Compactness and sequential completeness in some spaces of operators Let $$X$$ X be a completely regular Hausdorff space and $$C_b(X)$$ C b ( X ) be the Banach lattice of all real-valued bounded continuous functions on $$X$$ X , endowed with the strict topologies $$\beta _\sigma ,$$ β σ , $$\beta _\tau$$ β τ and $$\beta _t$$ β t . Let $$\mathcal{L}_{\beta _z,\xi }(C_b(X),E)$$ L β z , ξ ( C b ( X ) , E ) $$(z=\sigma ,\tau ,t)$$ ( z = σ , τ , t ) stand for the space of all $$(\beta _z,\xi )$$ ( β z , ξ ) -continuous linear operators from $$C_b(X)$$ C b ( X ) to a locally convex Hausdorff space $$(E,\xi ),$$ ( E , ξ ) , provided with the topology $$\mathcal{T}_s$$ T s of simple convergence. We characterize relative $$\mathcal{T}_s$$ T s -compactness in $$\mathcal{L}_{\beta _z,\xi }(C_b(X),E)$$ L β z , ξ ( C b ( X ) , E ) in terms of the representing Baire vector measures. It is shown that if $$(E,\xi )$$ ( E , ξ ) is sequentially complete, then the spaces $$(\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)$$ ( L β z , ξ ( C b ( X ) , E ) , T s ) are sequentially complete whenever $$z=\sigma$$ z = σ ; $$z=\tau$$ z = τ and $$X$$ X is paracompact; $$z=t$$ z = t and $$X$$ X is paracompact and Čech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on $$C_b(X)$$ C b ( X ) is given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Compactness and sequential completeness in some spaces of operators

, Volume 18 (2) – Jul 3, 2013
15 pages

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Publisher
Springer Journals
Copyright
Copyright © 2013 by The Author(s)
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0248-2
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

PositivitySpringer Journals

Published: Jul 3, 2013

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