If you have an analytic semigroup (generated by a so-called sectorial operator), then the answer is well-known and yes, even convergence rates are possible. This and generalizations were proved by many, see for example the summary and references in
In general it is possible to obtain convergence rates for "nice" initial values. A nice summary of the existing results and vast generalizations are available in this paper by Alexander Gomilko and Yuri Tomilov.
ADDED: After discussion with Delio, let me mention that the result (in Hilbert spaces) is implicitely contained in
H. FUJITA & A. MIZUTANI, "On the finite element method for parabolic equations. I; Approximation of holomorphic semi-groups," J. Math. Soc. Japan, v. 28, 1976, pp. 749-771.