On the nuclearity of integral operators

On the nuclearity of integral operators Let X be a nonempty measurable subset of $$\mathbb{R}^m$$ and consider the restriction of the usual Lebesgue measure σ of $$\mathbb{R}^m$$ to X. Under the assumption that the intersection of X with every open ball of $$\mathbb{R}^m$$ has positive measure, we find necessary and sufficient conditions on a L 2(X)-positive definite kernel $$K : X \times X \rightarrow \mathbb{C}$$ in order that the associated integral operator $$\mathcal {K} : L^2(X) \rightarrow L^2(X)$$ be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of $$L^2(X \times X)$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On the nuclearity of integral operators

Loading next page...
 
/lp/springer_journal/on-the-nuclearity-of-integral-operators-rPMRpjTu6V
Publisher
Birkhäuser-Verlag
Copyright
Copyright © 2008 by Birkhäuser Verlag Basel/Switzerland
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/s11117-008-2240-9
Publisher site
See Article on Publisher Site

Abstract

Let X be a nonempty measurable subset of $$\mathbb{R}^m$$ and consider the restriction of the usual Lebesgue measure σ of $$\mathbb{R}^m$$ to X. Under the assumption that the intersection of X with every open ball of $$\mathbb{R}^m$$ has positive measure, we find necessary and sufficient conditions on a L 2(X)-positive definite kernel $$K : X \times X \rightarrow \mathbb{C}$$ in order that the associated integral operator $$\mathcal {K} : L^2(X) \rightarrow L^2(X)$$ be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of $$L^2(X \times X)$$ .

Journal

PositivitySpringer Journals

Published: Oct 28, 2008

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 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.

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