On $m$-isometric semigroups, and $2$-isometric cogenerators

TL;DR

This paper characterizes generators of m-isometric semigroups using a Lumer-Phillips type approach and constructs a functional model for 2-isometric semigroups with analytic cogenerators, revealing new examples and limitations.

Contribution

It provides a new characterization of m-isometric semigroup generators and develops a functional model for 2-isometric semigroups with analytic cogenerators.

Findings

Characterization of m-isometric semigroup generators via m-skew-symmetry.

Construction of a functional model for 2-isometric semigroups.

Existence of 2-skew-symmetric operators not generating semigroups.

Abstract

It is known that a $C_0$-semigroup of Hilbert space operators is $m$-isometric if and only if its generator satisfies a certain condition, which we choose to call $m$-skew-symmetry. This paper contains two main results: We provide a Lumer--Phillips type characterization of generators of $m$-isometric semigroups. This is based on the simple observation that $m$-isometric semigroups are quasicontractive. We also characterize cogenerators of $2$-isometric semigroups. To this end, our main strategy is to construct a functional model for $2$-isometric semigroups with analytic cogenerators. The functional model yields numerous simple examples of $2$-isometric semigroups, but also allows for the construction of a closed, densely defined, $2$-skew-symmetric operator which is not a semigroup generator.