Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras

Spectral theoryis an important part of functional analysis.It has numerousapp- cations in many parts of mathematics and physics including matrix theory, fu- tion theory, complex analysis, di?erential and integral equations, control theory and quantum physics. In recent years, spectral theory has witnessed an explosive development. There are many types of spectra, both for one or several commuting operators, with important applications, for example the approximate point spectrum, Taylor spectrum, local spectrum, essential spectrum, etc. The present monograph is an attempt to organize the available material most of which exists only in the form of research papers scattered throughout the literature. The aim is to present a survey of results concerning various types of spectra in a uni?ed, axiomatic way. The central unifying notion is that of a regularity,which in a Banach algebra isasubsetofelementsthatareconsideredtobe“nice”.AregularityRinaBanach algebraA de?nes the corresponding spectrum ? (a)={?? C : a??/ ? R} in R the same wayas the ordinaryspectrum is de?ned by means of invertible elements, ?(a)={?? C : a??/ ? Inv(A)}. Axioms of a regularity are chosen in such a way that there are many natural interesting classes satisfying them. At the same time they are strong enough for non-trivial consequences, for example the spectral mapping theorem. Spectra ofn-tuples ofcommuting elements ofa Banachalgebraaredescribed similarly by means of a notion of joint regularity. This notion is closely related to ? the axiomatic spectral theory of Zelazko and S lodkowski.