Survey of last ten years of work done in India in some selected areas of functional analysis and operator theory

Abstract

Research in functional analysis and operator theory over the last ten years in India has broadly been under the following areas: 1. Hilbert space operator theory and certain special subsets of ℂ2 and ℂ3 where strong interaction between complex analysis and operator theory has been developed resulting in important discoveries in complex analysis through Hilbert space methods. 2. Hermitian holomorphic vector bundles and their curvatures. This has intimate connection with Hilbert modules over function algebras. A Hilbert module over a function algebra is a Hilbert space along with a bounded operator T or a commuting tuple of bounded operators (T1; T2; ... ; Tn): That is how operator theory enters this study. 3. Multivariable operator theory involving dilation of commuting contractions of certain special types. This involves the study of the submodule as well as the quotient module of the kind of Hilbert module mentioned above. This also includes important work on weighted shifts. 4. Operator theory motivated by mathematical physics. This includes trace formulae of Krein and Koplienko and their multivariable generalizations. Perturbation of self-adjoint operators is the main theme of study here. 5. From the seminal work from the 60’s by S. Kakutani, J. Lindenstrauss, G. Choquet and T. Ando, the study of geometric aspects of Banach spaces (the so called isometric theory) emerged as an important area. Indian researchers contributed both to the abstract study as well as towards understanding the structure of specific function spaces and spaces of operators. As the descriptions above show, operator theory has a great deal of relationship to several other branches of mathematics - geometry and complex analysis being perhaps the closest. This interaction is one source of pleasure that we shall try to bring out in this article. There has been a significant amount of excellent work done on operator algebras including Hilbert C*-modules and in matrix theory in India in recent times. Since there would be separate articles on those, this article does not include them.