Analytic multifunctions and operator theory

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My primary research interest is in the theory of analytic multifunctions and its applications in operator theory. The main objects of my research are to use the method of analytic multifunctions to the other domains.   Some of the problems I will look at are the following: (a) Spectral variation problems for algebraic elements in Banach algebra. Find the best possible estimate for Hausdorff distance and the other distances between the spectra of two algebraic elements. (b) I will continue to study the spectrum of a closed interpolated operator and the constancy problem of the spectrum. I will also study the particular case of the multifunctions induced by the semi-groups of partial differential operators and find the applications in the spectral theory of semi-groups. (c) Study the analytic multifunctions on the disk. Extend some well-known inequalities from complex analysis and potential theory to analytic multifunctions and then find the applications back to classic complex analysis. (d) Study the orthogonality between the range and kernel of the elementary operators in Hilbert space and in an abstract Banach algebra. Extend Putnam-Fuglede theorem and second degree Putnam -Fuglede theorem to the non-normal operators for elementary operators. The techniques involved come from the theory of analytic multifunctions, functional analysis, complex analysis,  Banach algebra, spectral theory, and operator theory.