Positive integral operators in unbounded domains

We study positive integral operators K in L2(R) with continuous kernel k(x, y). We show that if k(x, x) ∈ L1(R) the operator is compact and Hilbert–Schmidt. If in addition k(x, x) → 0 as |x| →∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and K is trace class. Replacing the first assumption by the stronger k1/2(x, x) ∈ L1(R) then k ∈ L1(R2) and the bilinear series converges also in L1. Sharp norm bounds are obtained and Mercer’s theorem is derived as a special case.  2004 Elsevier Inc. All rights reserved.