Analytic multifunctions and operator theory

我的主要研究兴趣是解析多函数理论及其在算子理论中的应用。本研究的主要目的是将多函数分析方法应用于其他领域。我将讨论的一些问题如下：(a)巴拿赫代数中代数元素的谱变化问题。找出两个代数元素光谱之间的豪斯多夫距离和其他距离的最佳估计。(b)我将继续研究闭插值算子的谱和谱的常数问题。本文还将研究由偏微分算子半群引起的泛函的特殊情况，并探讨其在半群谱理论中的应用。(c)研究磁盘上的分析多功能。将复分析和势理论中一些著名的不等式推广到解析泛函中，并将其应用到经典复分析中。(d)研究Hilbert空间和抽象Banach代数中初等算子的值域与核的正交性。将Putnam-Fuglede定理和二次Putnam-Fuglede定理推广到初等算子的非正规算子。所涉及的技术来自解析多函数理论、泛函分析、复分析、巴拿赫代数、谱理论和算子理论。

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.