On the characterization of eventually norm continuous semigroups in Hilbert spaces

We give a characterization of eventually norm continuous semigroups on Hilbert spaces in terms of its resolvent. It is an old problem to characterize eventually norm continuous semigroups through growth conditions for their resolvent. Beside the Hille–Yosida theorem for strongly continuous semigroups there are known characterizations for nilpotent (see [4, §6.10]), differentiable, eventually differentiable and analytic semigroups (see [3, Chapt.2.4, 2.5], [2, A-II]). In general Banach spaces a similar result for eventually norm continuous semigroups still seems to be unknown. However, recently P. You [5] showed that for semigroups in Hilbert spaces the norm continuity for t > 0 is equivalent to the decay to zero of its resolvent along some imaginary axis. The proof given in [5] is quite technical. It combines the complex inversion formula for the Laplace transform and a transformation of the integration path using the fact that the spectrum of the generator is “small”, i.e. bounded on imaginary axes. The aim of this paper is to give a much simpler and more straight forward proof of this result. Moreover, as an easy consequence we also obtain a characterization of semigroups which are norm continuous for t > t0. This result is of particular interest since for eventually norm continuous semigroups the spectral and the growth bounds coincide. Therefore the asymptotic behaviour of solutions of the associated Cauchy problem is determined by the location of the spectrum of the generator. For a more detailed analysis of the asymptotic behavior of strongly continuous semigroups we refer to [2, A–III, IV]. Theorem 1 [5, Thm.1]. Let A be the generator of a strongly continuous, exponentially stable semigroup (T (t))t≥0 on a Hilbert space H. Then the following conditions are equivalent. (a) (T (t))t≥0 is norm continuous for t > 0. (b) lim R3μ→±∞ ‖R(iμ,A)‖ = 0. Assertion (a)⇒(b) is well known, see e.g. [3, Chapt.2, Cor.3.7] but for the sake of completeness and to prove Theorem 4 we add the following result. In particular, it implies (a)⇒(b) if we choose F = L(H), S(t) = T (t) and μ0 = 0. 1991 Mathematics Subject Classification, Primary 47D03.