An Interesting Class of Operators with Unusual Schatten–von Neumann Behavior

We consider the class of integral operators Qφ on L (R+) of the form (Qφf)(x) = ∫ ∞ 0 φ(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 Sp, 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 Sp, 0 < p ≤ 1, the situation is more complicated. Although we give various necessary conditions and sufficient conditions relating to Qφ ∈ Sp 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 L and L modulus of continuity, respectively, with a rather small gap in between. A second cut-off occurs at p = 1/2: if φ is sufficiently smooth and decays reasonably fast, then Qφ belongs to the weak Schatten–von Neumann class S1/2,∞, but never to S1/2 unless φ = 0. We also obtain results for related families of operators acting on L(R) and l(Z). We further study operations acting on bounded linear operators on L(R) related to the class of operators Qφ. 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 Qφ.