Two Classes Of Operators And Local Spectral Theory

Spectral theory is an important part of functional analysis. It has numerous applications in many parts of mathematics and physics including matrix theory, func-tion theory, complex analysis, differential and integral equations, control theory and quantum physics.The classical study of the spectral structure of an operator has recently been enriched by the development of some powerful new methods for the local analysis of the spectrum. In particular, the deep interaction between the Fredholm theory and local spectral theory becomes evident when one consider the so called single-valued extension property(SVEP).In this paper, we mainly discuss two generalizations of operators of Kato type—the operators which admit a generalized Kato decomposition and operators which have topological uniform descent on Banach spaces by means of local spectral theory. Our main results can be divided into two parts.One is for operators which admit a generalized Kato decomposition. We com-pletely classify all of the characterizations of SVEP at a point of operators of Kato type and their adjoints so far as we know into two classes:one can be extended to operators which admit a generalized Kato decomposition and their adjoints, and the other one can not be extended. In particular, we characterize the SVEP at a point for operators which admit a generalized Kato decomposition and their adjoints by means of the accumulation points of the approximate point spectrum, as well as of the surjectivity spectrum. And we give counterexamples to show why some charac-terizations can not be extended. We then study the components of generalized Kato resolvent set of an operator:we show the constancy of certain subspace valued map-pings on these components; By means of the constancy and, in the case of operators which admit a generalized Kato decomposition, the equivalences of the SVEP at a point, we obtain a classification of these components. According to the classification, we get some useful information on the fine structure of the spectrum.The other is for operators which have topological uniform descent. We show that, all of the characterizations of SVEP at a point of operators of Kato type and their adjoints so far as we know, can be extended to operators which have topological uniform descent and their adjoints. We also study the components of topological uni-form descent resolvent set of an operator:we show the constancy of certain subspace valued mappings on these components; By means of the constancy and, in the case of operators which have topological uniform descent, the equivalences of the SVEP at a point, we obtain a classification of these components; According to this classification, we obtain more useful information on the fine structure of the spectrum.The methods which we use to deal with the two classes of operators are com-pletely different, and they are also different from the one for operators of Kato type.It is worth to mention that, when we discuss the two classes of operators, we get such a deep result:T is of Kato type if and only if T admits a generalized Kato decomposition and T has topological uniform descent. Moreover, by means of this result, we obtain a new spectral structural diagram of an operator.