## On the asymptotic behavior of spectral characteristics of an integral operator with difference kernel on expanding regions.(English. Russian original)Zbl 0635.47006

Sov. Math., Dokl. 33, 400-403 (1986); translation from Dokl. Akad. Nauk SSSR, 287, 529-532 (1986).
The authors consider an integral operator $$A_ t$$ with difference kernel of the form $$t^ d{\mathcal A}(t(x-y))$$, where $${\mathcal A}:R^ d\to C$$ is a functional in the Schwartz class $${\mathcal S}(R^ d)$$. The asymptotic behavior of the trace tr$$f(A_ t)$$ of a function of $$A_ t$$ is given. The case $$d=1$$, which is degenerate in certain respects, is written out separately. The obtained results admit a natural generalization to an operator $$\tilde A_ t$$ with a kernel of the form $$t^ d{\mathcal A}(t(x- y),x,y)$$. The asymptotic behavior of tr$$f(\tilde A_ t)$$ was recently got by H. Widom but the technique, which the authors used is fairly elementary. This paper contains simple formulas for the resolvent.
Reviewer: L.Hacia

### MSC:

 47A10 Spectrum, resolvent 47B38 Linear operators on function spaces (general) 47Gxx Integral, integro-differential, and pseudodifferential operators

### Keywords:

integral operator; difference kernel; Schwartz class