Research Projects

Toeplitz operators on Hardy space of unit disc is one of the most studied and well-understood objects in the theory of bounded linear operators. It serves as a bridge between function theory, index theory C* algebra and Operator theory. Otto Toeplitz first introduced these operators in their matrix form in the early part of the 20th century. However, the seminal work of P. Hamos, A. Brown, D. Sarason, and R. Douglas (to name a few) in the latter half of the century pushed the study of Toeplitz operators deep into modern operator theory. Although the theory of Toeplitz operators is old and well-studied in one variable case, several variable cases remain widely open and less understood. The primary reason for extending the ideas in one complex variable to several variables is the need for well-suited generalisations. In this project, we propose some natural questions for Teoplitz operators in several variables. We want to provide a complete characterisation of Toeplitz operators satisfying T^*T-TT^*greater than or equal to 0 (hyponormal), TT^*T = T (partial-isometry) and related (similar) *identities on (weighted) Bergman space, Hardy space, of the unit Ball. Moreover, we also want to understand the behaviour of such operators in some general K-invariant domains. Even though these problems, as stated in the project, seem domain-specific, the underlying structure remains the same in different complex domains. In particular, through this project, we hope to find a unified approach to improve understanding of these structures.