$C_0$-semigroups of $m$-isometries on Hilbert spaces

TL;DR

This paper characterizes $C_0$-semigroups of $m$-isometries on Hilbert spaces through polynomial conditions on norms, explores their properties on weighted spaces, and provides criteria for embedding certain $2$-isometries into semigroups.

Contribution

It offers a novel characterization of $m$-isometric semigroups via polynomial norm conditions and analyzes their structure and embedding properties.

Findings

Characterization of $m$-isometric semigroups using polynomial degree conditions.

Conditions on the infinitesimal generator and cogenerator for $m$-isometries.

Embedding criteria for non-unitary $2$-isometries satisfying the kernel condition.

Abstract

Let $\{T(t)\}_{t\ge 0}$ be a $C_0$-semigroup on a separable Hilbert space $H$. We characterize that $T(t)$ is an $m$-isometry for every $t$ in terms that the mapping $t\in \Bbb R^+ \rightarrow \|T(t)x\|^2$ is a polynomial of degree less than $m$ for each $x\in H$. This fact is used to study $m$-isometric right translation semigroup on weighted $L^p$-spaces. We characterize the above property in terms of conditions on the infinitesimal generator operator or in terms of the cogenerator operator of $\{ T(t)\}_{t\geq 0}$. Moreover, we prove that a non-unitary $2$-isometry on a Hilbert space satisfying the kernel condition, that is, $$ T^*T(KerT^*)\subset KerT^*\;, $$ then $T$ can be embedded into a $C_0$-semigroup if and only if $dim (KerT^*)=\infty$.