C0-semigroups of m-isometries on Hilbert spaces

Abstract

Let { T ( t ) } t ≥ 0 be a C 0 -semigroup on a separable Hilbert space H . We show that T ( t ) is an m -isometry for any t if and only if the mapping t ∈ R + → ‖ T ( t ) x ‖ 2 for each x ∈ H is a polynomial of degree at most m . This property is used to study m -isometric right translation semigroup on weighted L p -spaces. We also provide alternative characterizations of the above property by imposing conditions on the infinitesimal generator operator and on the cogenerator operator of { T ( t ) } t ≥ 0 . Moreover, we prove that a non-unitary 2-isometry T on a Hilbert space satisfying the kernel condition, that is, T ⁎ T ( K e r T ⁎ ) ⊂ K e r T ⁎ , can be embedded into a C 0 -semigroup if and only if d i m ( K e r T ⁎ ) = ∞ .