The spectral mapping theorem for joint approximate point spectrum

Abstract

The spectral mapping theorem for joint approximate point spectrum is proved when A is an «-tuple of commuting operators on a Banach space and ƒ is any w-tuple of rational functions for which f (A) is defined. The purpose of the paper is to show how properties of parts of the joint spectrum can be obtained by use of spaces of sequences of vectors. The method (first used for general Banach spaces in [4]) provides very simple proofs of some known results ; but the main result is believed to be new. Throughout the paper, we deal with n-tuples of bounded operators on a Banach space 2£ 9 whose vectors are denoted by x, y, • • •. It will be convenient to use a symbol such as A to denote an «-tuple of operators and a symbol such as A to denote an «-tuple of complex numbers: A = (Al9 • • • , An)9 A=(Al5 • • • , AJ, 0=(0 , • • • , 0). The one-sided spectra of interest in this paper are not defined in terms of existence of Banachalgebra inverses, the context of the papers of Robin Harte ([5], [6]); they are the somewhat different spectra of the following definition. (However, in the special case that 2£ is a Hilbert space, our result is a corollary of Harte's.) DEFINITION. We say 0eov(A) in case there exists a nonzero xe& such that Ax=0 (i.e., AjX=0 for j=l, • • • , « ) . We say 0 e av(A) in case for every £>0 there exists a unit vector xeSC such that \Ax\<& (i.e., \\AjX <e for j=l, • • • , n). We say 0 G ao(Al9 • • • , An) in case 0 e av(A*9 • • • , A*), where A* denotes the Banachspace adjoint of A$. We say X G av(A) in case 0 G ap(A—A), and similarly for aV9 ao. The set av(A) is called the 'point spectrum' of the n-tuple A; the set ov(A)9 its 'approximate point spectrum' or 'left approximate spectrum' ; the set ao(A), its 'approximate defect spectrum' or 'right approximate spectrum'. AMS (MOS) subject classifications (1970). Primary 47A10; Secondary 47A20, 47A60.