# Improving integrability via absolute summability: a general version of Diestel’s Theorem

Improving integrability via absolute summability: a general version of Diestel’s Theorem A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the $$(p,\sigma )$$ ( p , σ ) -absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces—including C(K), $$L^p$$ L p and Hilbert spaces—and operators—p-summing, (q, p)-summing and p-approximable operators. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Improving integrability via absolute summability: a general version of Diestel’s Theorem

, Volume 20 (2) – Sep 22, 2015
15 pages

/lp/springer_journal/improving-integrability-via-absolute-summability-a-general-version-of-wBLZj9w0nY
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/s11117-015-0361-5
Publisher site
See Article on Publisher Site

### Abstract

A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the $$(p,\sigma )$$ ( p , σ ) -absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces—including C(K), $$L^p$$ L p and Hilbert spaces—and operators—p-summing, (q, p)-summing and p-approximable operators.

### Journal

PositivitySpringer Journals

Published: Sep 22, 2015

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