An Interesting Class of Operators with unusual Schatten-von Neumann behavior

We consider the class of integral operators $Q_\f$ on $L^2(\R_+)$ of the form $(Q_\f f)(x)=\int_0^\be\f (\max\{x,y\})f(y)dy$. We discuss necessary and sufficient conditions on $\phi$ to insure that $Q_{\phi}$ is bounded, compact, or in the Schatten-von Neumann class $\bS_p$, $1<p<\infty$. We also give necessary and sufficient conditions for $Q_{\phi}$ to be a finite rank operator. However, there is a kind of cut-off at $p=1$, and for membership in $\bS_{p}$, $0<p\leq1$, the situation is more complicated. Although we give various necessary conditions and sufficient conditions relating to $Q_{\phi}\in\bS_{p}$ 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 $L^1$ and $L^2$ 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 $\wS{1/2}$, but never to $\bS_{1/2}$ unless $\f=0$. We also obtain results for related families of operators acting on $L^2(\R)$ and $\ell^2(\Z)$. We further study operations acting on bounded linear operators on $L^{2}(\R^{+})$ related to the class of operators $Q_\f$. In particular we study Schur multipliers given by functions of the form $\phi(\max\{x,y\}) $ and we study properties of the averaging projection (Hilbert-Schmidt projection) onto the operators of the form $Q_\f$.