Remarks on normal operators on Banach spaces (Q1097490)

A closed densely defined operator T on a Banach space X is called normal, iff \(T\in [C^ 0({\hat {\mathbb{C}}})]\), i.e. there is a homomorphism (functional calculus) \(f\to f(T)\) of the continuous functions on \({\hat {\mathbb{C}}}:={\mathbb{C}}\cup \{\infty \}\) into the bounded linear operators with \(\| f(T)\| \leq \sup \{| f(z)|:z\in {\hat {\mathbb{C}}}\}\) and which extends the analytic calculus. It is shown that this definition agrees with that of \textit{T. W. Palmer} [Trans. Amer. Math. Soc. 133, 385- 414 (1968; Zbl 0169.169)]. The functional calculus yields simple proofs of properties which are well-known for normal operators on Hilbert spaces. In the second part the methods are applied to semi-groups of normal operators resp. cosine operator functions.