2001/03/05 by А. Б. Александров, A. B. Aleksandrov, Aleksandrov, A. B. +9 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math.CA #math.CV #math.FA
paper · pdf · doi:10.48550/arxiv.math/0103028
87 pages
arxiv created 2001/03/05 · arxiv updated 2009/11/30
We consider the class of integral operators Q_\f on L2(\R+) of the form (Q_\f f)(x)=∫0^\be\f (max\x,y\)f(y)dy. We discuss necessary and sufficient conditions on ϕ to insure that Qϕ is bounded, compact, or in the Schatten-von Neumann class \bSp, 1<p<∞. We also give necessary and sufficient conditions for Qϕ to be a finite rank operator. However, there is a kind of cut-off at p=1, and for membership in \bSp, 0<p≤1, the situation is more complicated. Although we give various necessary conditions and sufficient conditions relating to Qϕ∈\bSp in that range, we do not have necessary and sufficient conditions. In the most important case p=1, we have a necessary condition and a sufficient condition, using L1 and L2 modulus of continuity, respectively, with a rather small gap in between. A second cut-off occurs at p=1/2: if \f is sufficiently smooth and decays reasonably fast, then \qf belongs to the weak Schatten-von Neumann class \wS1/2, but never to \bS1/2 unless \f=0. We also obtain results for related families of operators acting on L2(\R) and ℓ2(\Z). We further study operations acting on bounded linear operators on L2(\R+) related to the class of operators Q_\f. In particular we study Schur multipliers given by functions of the form ϕ(max\x,y\) and we study properties of the averaging projection (Hilbert-Schmidt projection) onto the operators of the form Q_\f.