# 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

/lp/springer_journal/compactness-and-sequential-completeness-in-some-spaces-of-operators-50wFd0OBXB
Publisher
Springer Journals
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations