Abstract
Let E be a complex Banach space and T a bounded linear operator. If σw(T) is the set of all spectral points of T which are not poles of the resolvent and rw(T) its radius, we give formulas for the moduli of all poles lying outside the circle with radius rw(T). We further prove that\(r_w (T) = \mathop {\lim }\limits_{k \to \infty } [\pi (T^k )]^{1/k} \), where π(T)=inf(∥T−F∥∶F(T−F)=(T−F)F=0,F∈F and F is the Caradus class of operators with rational resolvent. Finally we give an estimation of rw(S+T) and rw(S·T).
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Wacker, H.D. Formeln für Pole der Resolvente und den Radius eines wesentlichen Spektrums. Manuscripta Math 54, 249–259 (1986). https://doi.org/10.1007/BF01171336
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DOI: https://doi.org/10.1007/BF01171336