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arXiv · 1809.00664

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

Eskil Rydhe
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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.

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Eskil Rydhe. 2018-09-03. On $m$-isometric semigroups, and $2$-isometric cogenerators. https://doi.org/10.1007/s00020-019-2524-4

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