# 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

, Volume 13 (3) – Oct 28, 2008
23 pages

/lp/springer_journal/on-the-nuclearity-of-integral-operators-rPMRpjTu6V
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/s11117-008-2240-9
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

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