The distance from the Apostol spectrum

Abstract

If T T is an s-regular operator in a Banach space (i.e. T T has closed range and N ( T ) ⊂ R ∞ ( T ) N(T)\subset R^{\infty }(T) ) and γ ( T ) \gamma (T) is the Kato reduced minimum modulus, then lim n → ∞ γ ( T n ) 1 / n = sup { r : T − λ i s s - r e g u l a r f o r | λ | > r } . \begin{equation*}\lim _{n\to \infty }\gamma (T^{n})^{1/n}=\sup \{r: T-\lambda { \operatorname {is s-regular for }}|\lambda |>r\}. \end{equation*}