We consider the class of integral operators Q_\f on L2(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 ϕ to insure that Qϕ is bounded, compact,
or in the Schatten-von Neumann class \bS_p, 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
\bS_{p}, 0<p≤1, 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 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 \wS{1/2}, but never to
\bS_{1/2} 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.Comment: 87 page