C₀-semigroups of m-isometries on Hilbert spaces

2018/10/17 by Bermudez, T., Bonilla, A., Zaway, H.
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1810.07494

Abstract

Let \T(t)\_{t≥ 0} be a C₀-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∈ \Bbb R⁺ → ‖T(t)x‖² is a polynomial of degree less than m for each x∈ 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≥ 0}. Moreover, we prove that a non-unitary 2-isometry on a Hilbert space satisfying the kernel condition, that is, T^*T(KerT^*)⊂ KerT^* , then T can be embedded into a C₀-semigroup if and only if dim (KerT^*)=∞.

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• C₀-semigroups of 2-isometries and Dirichlet spaces
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• Some examples of m-isometries
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