Integral operators of trace class (Q5917708)

Given a measurable square integrable kernel \(K(x, \xi)\), \(-\infty\leq a< x\), \(\xi< b\leq +\infty\), the problem of finding sufficient conditions for the associated integral operator \(K\) on \(L^2(a, b)\) to be of trace class has been studied, among others, by Hille and Tamarkin, Gohberg and Krein, when \(-\infty< a< b< +\infty\), and by \textit{W. F. Stinespring} [J. Reine Angew. Math. 200, 200-207 (1958; Zbl 0095.095)], \textit{W. P. Kamp}, \textit{R. A. Lorentz} and \textit{P. A. Rejto} [ibid. 393, 1-20 (1989; Zbl 0659.45013)], \textit{J. Gonzalez-Barrios} and \textit{R. M. Dudley} [Proc. Am. Math. Soc. 118, 175-180 (1993; Zbl 0791.47023)], when \(a= -\infty\), \(b= +\infty\). The results in the second cited paper are generalizations of those in the first cited one, but the proofs are very technical. It is the object of this work, to show that imposing essentially the same conditions on the kernel \(K(x, \xi)\) we can prove that the associated integral operator is of trace class, utilizing very elementary considerations.