Abstract. We shall show in this paper a quite useful formula connecting the essential minimum modulus and minimum modulus of any linear bounded operator defined on a separable Hilbert space. Since this formula does not depend on the structure of Hilbert space, this result enables us to define the essential minimum modulus of linear operators in the more general context of Banach spaces. The connection between our definition and that given by Zemánek [Geometric interpretation of the essential minimum modulus, Operator Theory: Adv. Appl., 6, (1982), 225-227] is discussed. Moreover, the notion of the essential surjectivity modulus and left (resp. right) essential minimum modulus on Banach spaces are also defined and will be studied in this paper. The asymptotic formula for the essential spectrum of a semi-Fredholm operator with index zero in terms of the left and right essential minimum moduli is proved.

DOI: 10.14232/actasm-014-538-8

AMS Subject Classification (1991): 47A53, 47A55, 47A60, 46B04

Keyword(s): Banach space, minimum modulus, surjectivity modulus, essential minimum modulus, Calkin algebra, essential spectrum, semi-Fredholm, index

Received May 14, 2014, and in final form January 17, 2016. (Registered under 38/2014.)