An Interesting Class of Operators with unusual Schatten-von Neumann behavior
| dc.creator | Aleksandrov, A. B. | |
| dc.creator | Janson, S. | |
| dc.creator | Peller, V. V. | |
| dc.creator | Rochberg, R. | |
| dc.date | 2001-03-05 | |
| dc.date.accessioned | 2026-07-07T04:40:30Z | |
| dc.date.available | 2026-07-07T04:40:30Z | |
| dc.description | 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 $ϕ$ to insure that $Q_ϕ$ is bounded, compact, or in the Schatten-von Neumann class $\bS_p$, $1<p<\infty$. We also give necessary and sufficient conditions for $Q_ϕ$ 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 $Q_ϕ\in\bS_{p}$ 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 $ϕ(\max\{x,y\}) $ and we study properties of the averaging projection (Hilbert-Schmidt projection) onto the operators of the form $Q_\f$. | |
| dc.description | 87 pages | |
| dc.identifier | https://arxiv.org/abs/math/0103028 | |
| dc.identifier | http://arxiv.org/abs/math/0103028 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61045 | |
| dc.subject | Functional Analysis | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Complex Variables | |
| dc.title | An Interesting Class of Operators with unusual Schatten-von Neumann behavior | |
| dc.type | text |