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 `(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.