Absolutely continuous operators on function spaces and vector measures

Absolutely continuous operators on function spaces and vector measures Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L 0(μ) such that $${L^\infty(\mu) \subset E \subset L^1(\mu)}$$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space $${(X, \xi)}$$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T m : L ∞(μ) → X. In particular, we characterize relatively compact sets $${\mathcal{M}}$$ in ca μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology $${\mathcal{T}_s}$$ of simple convergence in terms of the topological properties of the corresponding set $${\{T_m : m \in \mathcal{M}\}}$$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ∞(μ) → X. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Absolutely continuous operators on function spaces and vector measures

Positivity , Volume 17 (3) – Jun 26, 2012
Loading next page...
 
/lp/springer_journal/absolutely-continuous-operators-on-function-spaces-and-vector-measures-34udjz8vJq
Publisher
Springer Basel
Copyright
Copyright © 2012 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/s11117-012-0187-3
Publisher site
See Article on Publisher Site

Abstract

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L 0(μ) such that $${L^\infty(\mu) \subset E \subset L^1(\mu)}$$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space $${(X, \xi)}$$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T m : L ∞(μ) → X. In particular, we characterize relatively compact sets $${\mathcal{M}}$$ in ca μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology $${\mathcal{T}_s}$$ of simple convergence in terms of the topological properties of the corresponding set $${\{T_m : m \in \mathcal{M}\}}$$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ∞(μ) → X.

Journal

PositivitySpringer Journals

Published: Jun 26, 2012

References

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.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

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.

See the journals in your area

DeepDyve Freelancer

DeepDyve Pro

Price
FREE
$49/month

$360/year
Save searches from
Google Scholar,
PubMed
Create lists to
organize your research
Export lists, citations
Read DeepDyve articles
Abstract access only
Unlimited access to over
18 million full-text articles
Print
20 pages/month
PDF Discount
20% off