Spectral properties of operator logarithms (Q1434522)

Let \(S_{\omega}=\{ z \in \mathbb C : z \neq 0, | \text{arg}z | <\omega \}\), \(0<\omega\leq \pi\), and let \(A\) be a linear injective operator on a Banach space \(X\). Suppose that \(\sigma(A) \subset {\overline {S_{\omega}}}\) and \(\sup \{ \| \lambda R(\lambda,A) \| : \lambda \notin {\overline {S_{\omega^{\prime}}}} \} <\infty\) for all \(\omega<\omega^{\prime}<\pi\) (in short: \(A \in \text{Sect}(\omega)\)). The number \(\omega_A=\min\{ \omega : A \in \text{Sect}(\omega)\}\) is called the spectral angle of \(A\). An arbitrary linear operator \(B: X \to X\) is said to be a strip operator of height \(\omega\) (in short: \(B \in \text{Strip}(\omega)\)), if \(\sigma(B) \subset \{ z \in \mathbb{C} : | \text{Im} z| \leq \omega \}\) and \(\sup \{ \| R(\lambda, B)\| : | \text{Im} \lambda | \geq \omega^{\prime} \} <\infty\) for all \(\omega^{\prime}>\omega\). Let \(\omega_{st}(B)=\min \{ \omega \geq 0 : B \in \text{Strip}(\omega) \}\). The main theorem states that if \(A \in \text{Sect}(\omega)\) is an injective operator such that \(\log A \in \text{Strip}(\omega^{\prime})\), \(\omega^{\prime} \geq 0\), then \(A \in \text{Sect}(\omega^{\prime})\). In particular, \(\omega_{st}(\log A)=\omega_A\). Moreover, \(A\) is uniquely determined by its logarithm \(\log A\).