A Note on Logarithms of Normal Operators
Proceedings of the American Mathematical Society
All operators considered in this note are bounded and defined on a fixed Hubert space X. In  , C. R. Putnam has proved that if 77 is a positive definite selfadjoint operator and exp T = H, then ||f|| 2 In 2 implies that F is a selfadjoint operator. In Theorem 3 we prove that it is sufficient to assume that || T\\ <2ir in order that T be selfadjoint. This condition, already in the set of complex numbers, cannot be replaced by ||f|| ;£2ir without changing the conclusion. In Theorem 4 we prove
... hat ||F|| ^2t in a finite dimensional space implies that F is a normal operator. In Theorem 2 some conditions for a logarithm F of a normal operator N are derived. Assuming that the spectrum of N lies in the set (1) Q = he* | -air Û 0\ we prove that exp T = N and ||F|| <(1-a2/4:)ir imply that F is a normal operator. All these results are consequences of Theorem 1. Let X be a Hubert space and T: X-+X a bounded operator such that exp T=N is a normal operator. Then (2) where In X is the principal (or any) branch of the logarithmic function and E(X) is the spectral measure of N. The bounded operator W commutes with No and there exists a bounded and regular, positive definite selfadjoint operator Q such that (4) JFo = QrlWQ is a selfadjoint operator the spectrum of which belongs to the set of all integers (cf. [2, Theorem 4; 3, Lemma 2.2 and Theorem III]). Proof. The operator N0 defined by (3) is bounded and exp N0 = N.