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

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 $t\in {\mathbb{R}}^{+}\to \Vert T(t)x{\Vert }^2$ 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 $\{T(t){\}}_{t\ge 0}$. Moreover, we prove that a non-unitary $2$-isometry on a Hilbert space satisfying the kernel condition, that is, $$T^{\ast }T(Ker{T}^{\ast })\subset Ker{T}^{\ast },$$ then $T$ can be embedded into a $C_0$-semigroup if and only if $dim(Ker{T}^{\ast })=\infty$.

17 pages