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Mathematics : Functional Analysis
- Banach Space Theory :In Banach space theory the key areas of research are the following: (i) Approximation theory in infinite dimensional spaces with special emphasis on classical spaces. (ii) Isomorphic theory of separable Banach spaces, saturation and decomposition.
- Operator Spaces :The main emphasis is on operator space techniques in Abstract Harmonic Analysis.
- Operator Theory :The interest in this area, as represented by our department, is along the following two directions :
- Unbounded Subnormals The most outstanding example of an unbounded subnormal is the Creation Operator of the Quantum Mechanics. Our analysis of these operators is essentially based on the theory of Sectorial forms, a sophisticated tool from PDEs. In particular, one may combine the theory of sectorial forms with the spectral theory of unbounded subnormals to derive polynomial approximation results on certain unbounded regions.
- Operators Close to Isometries This is huge subclass of left-invertible operators which behave like isometries of Hilbert spaces. One may develop an axiomatic approach to these operators. Via this axiomatization, one may obtain the Beurling-type theorems for Bergman shift and Dirichlet shift in one stroke. Important examples of these operators include 2-hyperexpansive operators and Bergman-type operators. There is a transform which sends 2-hyperexpansive operators to Bergman-type operators. For instance, one may use this transform to obtain the Berger-Shaw theory for 2-hyperexpansive operators from the classical Berger-Shaw theory.
- Real and Complex Analysis, Nonlinear Analysis
- Harmonic Analysis