Differentiability and norm continuity

Summary

Section 6.1 is intended to give a characterization of the infinitely differentiable propagators of (ACP _n ) in Banach spaces, which depends only on the properties of $$. As a corollary, a concise sufficient condition is also presented.

Section 6.2 explores the characterization of the norm continuity (i.e., continuity in the uniform operator topology) for t > 0 of the propagators of (ACP _n ) in Hilbert spaces. Following a general discussion on Laplace transforms in this respect, we obtain a succinct characterization (Theorem 2.1).

In Section 6.3, we restrict to (ACP ₂) in a Banach space with A ₁ ∈ L(E); see also Section 2.5. We show that S ₀(t) or S ^′₁ (t) is norm continuous for t > 0 if and only if A ₀ is bounded. This leads to an interesting consequence for strongly continuous cosine operator functions or operator groups.

Section 6.4 is concerned with the operator matrix

$$$$

, where A is a positive self-adjoint operator in a Hilbert space and B subordinated to A in various ways. One can see that the semigroup generated by A _B (or $$) may possess norm continuity, differentiability, analyticity, or exponential stability, respectively, as B changes.