Level sets of the condition spectrum

D, Sukumar and Veeramani, S (2017) Level sets of the condition spectrum. Annals of Functional Analysis, 8 (3). pp. 314-328. ISSN 2008-8752

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Abstract

For 0 < ε ≤ 1 and an element a of a complex unital Banach algebra A, we prove the following two topological properties about the level sets of the condition spectrum. (1) If ε = 1, then the 1-level set of the condition spectrum of a has an empty interior unless a is a scalar multiple of the unity. (2) If 0 < ε < 1, then the ε-level set of the condition spectrum of a has an empty interior in the unbounded component of the resolvent set of a. Further, we show that, if the Banach space X is complex uniformly convex or if X* is complex uniformly convex, then, for any operator T acting on X, the level set of the ε-condition spectrum of T has an empty interior.

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