Cayley transform of the generator of a uniformly bounded C_0-semigroup of operators - Gomilko, A. M. (Ukrains’kyi Matematychnyi Zhurnal, 2004)
We consider the problem of estimates for the powers of the Cayley transform V = (А + I)(А - I)^-1 of the generator of a uniformly bounded C_0-semigroup of operators e^tA , t ≥ 0, that acts in a Hilbert space H. In particular, we establish the estimate _n N(||V^n||/(n + 1)) < ∞. We show that the estimate _n ∈ N ∥V^n∥ < ∞ is true in the following cases: (a) the semigroups e^tA and e^tA^−1 are uniformly bounded; (b) the semigroup etA uniformly bounded for t ≥ ∞ is analytic (in particular, if the generator of the semigroup is a bounded operator).