Research Projects

Normal operators and isometries are well understood using the spectral theorem and the Wold decomposition, respectively. An operator A is said to be quasinormal if A commutes with A*A, or equivalently, U|A|=|A|U, where A=U|A| is the polar decomposition of A. Clearly, normal operators and isometries are examples of quasinormal operators. Quasinormal operators retain key spectral properties despite being more general than normal operators. They arise naturally in moment problems, function theory, and dilation theory, making them central to the structural study of non-normal operators and deeply connected to many core themes in operator theory. Brown characterized all quasinormal operators. In fact, he showed that A is quasinormal if and only if A is unitarily equivalent to $(M_z\otimes P)\oplus N,$ where M_z denotes the multiplication by z on the scalar valued Hardy space H^2(D), P is a positive operator and N is a normal operator. In the case where A is an isometry, we have P=I, the identity operator and N as a unitary operator (Wold decomposition). It is natural to wonder what can be said about pairs of commuting quasinormal operators. That is, what is the analogue of Brown's result? Berger, Coburn, and Lebow partially addressed this question by describing the structure of a pair of commuting isometries (V_1,V_2). This project aims to derive the structure for pairs of commuting quasinormal operators, that is, an analogue of Brown's result. An explicit structure is crucial, for example, in determining the Taylor joint spectrum. Hence, we also aim to study the Taylor joint spectrum for pairs of commuting quasinormal operators. In my thesis, we established results on the Taylor joint spectrum for certain pairs of commuting isometries. Having understood the structure in multi-operator settings, a natural progression is to explore analogous questions for commuting semigroups of quasinormal operators. Semigroups offer a powerful framework for studying dynamical systems, evolution equations, and system dynamics, with applications in control theory, quantum mechanics, and signal processing. Embry-Wardrop gave an explicit structure for semigroups of quasinormal operators. It would be interesting to investigate the structure of pairs of commuting semigroups of quasinormal operators (In my thesis, we established the structure of pairs of commuting semigroups of isometries). Of course, the above problems are equally meaningful for tuples, not just pairs, and we intend to address these as well. Similar to the case of a single contraction, it is well known that a contractive semigroup can be dilated to an isometric semigroup. Building on this and the commutant of the right-shift-semigroup obtained in my thesis, we recently established a Sarason-type commutant lifting theorem in the semigroup framework. In future work, we plan to investigate dilation and establish a commutant lifting theorem for families of commuting contractive semigroups.