On m-Isometric Semigroups, and 2-Isometric Cogenerators

It is known that a C0-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 non-unitary 2-isometric semigroups, but also allows for the construction of a closed, densely defined, 2-skew-symmetric operator which is not a semigroup generator.