Date posted: July 2021.

Normally I write here about very recent pieces of exciting work, but this is a post about a slightly older work because there are unfulfilled potential there which can make excellent future projects. It places the open symmetrized bidisc front and center of a discussion about Toeplitz operators and unravels a wealth of information and strong results about it. As is well known, the symmetrized bidisc behaves considerably different than the bidisc, so results about Toeplitz operators are not easily discerned at first.

This work would not have happened if Misra, Shyam Roy and Zhang did not discover the Hardy space of the symmetrized bidisc in Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc . We find three isomorphic copies of the Hardy space as \( D^2 \)-contractive Hilbert modules. This allows us to embed the Hardy space inside \( L^2 \) of the boundary with respect to a natural measure which in turn allows us to define Toeplitz operators. There is an algebraic characterization of Toeplitz-ness similar to what Brown and Halmos discovered for classical Toeplitz operators back in 1963 for the open unit disc in the complex plane. Our work goes on to study the asymptotic behavior, commutant lifting theorem, analytic Toeplitz operators and dual Toeplitz operators. The work provokes several natural questions. What can be said about the spectrum of a Toeplitz operator in this context? What about index theory? Perhaps more fundamentally, is there a class of inhomogeneous domains for which there is a good theory of Toeplitz operators?

We were glad to set the scene for the study of such questions. This paper is available at International Mathematics Research Notices (IMRN), Volume 2021, Issue 11, June 2021, Pages 8763 - 8805 and on arXiv.

Date posted: May 2021.

The study of a pair \( (V_1, V_2) \) of commuting isometries. This is a classical theme. We managed to shine new light on it by using the defect operator of a pair of commuting isometries which is defined as $$C(V_1, V_2) = I - V_1V_1^* - V_2V_2 ^* + V_1V_2V_2^* V_1^* . $$ In the cases when the defect operator is zero or positive or negative, or the difference of two mutually orthogonal projections adding up to the kerel of \( (V_1V_2)^* \), we write down structure theorems for \( (V_1, V_2) \). The structure theorems allow us to compute the joint spectra in each of the cases above. Moreover, in each case, we also point out at which stage of the Koszul complex the exactness breaks. With \( (V_1, V_2) \), a pair of operator valued functions \( (\varphi_1, \varphi_2) \) can be canonically associated. In certain cases, the closure of the union of the joint spectra of \( (\varphi_1(z), \varphi_2(z)) \) is the same as the joint spectrum of \( (V_1, V_2) \) and in certain cases, it is smaller. All such cases are described. We are particularly happy to figure out the fundamental pair of commuting isometries with negative defect. We construct this pair of commuting isometries on the Hardy space of the bidisc (It has been known that the fundamental pair of commuting isometries with positive defect is the pair of multiplication operators by the coordinate functions on the Hardy space of the bidisc).

Date posted: May 2020.

The foliation of the symmetrized bidisc. This is the work with Anindya Biswas, who is my Ph.D. student now and Anwoy Maitra, who had just finished his Ph.D. with Gautam Bharali when the work was done. Anindya and I were trying to see how the automorphism group of the symmetric bidisc acts on itself. We figured out that the \( 3 \) (real) dimensional automorphism group of the symmetrized bidisc gives rise to orbits, all of which, except one, are \( 3 \) dimensional hypersurfaces. The exceptional one is an analytic disc. We thought that such complex manifolds must have been studied before and indeed, Isaev had studied in great detail \( 2 \) (complex) dimensional Kobayashi hyperbolic manifolds with \( 3 \) (real) dimensional automorphism groups. Isaev had classified them. So, the symmetrized bidisc must figure in his list. However, Isaev's list is long. At this point, it was getting clear that we need a genuine complex analyst. So, Anindya roped Anwoy in. Gradually, one discovered that a certain unbounded domain \( \mathcal D_1 \) in Isaev's list is biholomorphic to the symmetrized bidisc. The road to the biholomorphism is paved with various geometric insights about the symmetrized bidisc. Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of \( \mathcal D_1 \). Of particular interest is the fact that \( \mathcal D_1 \) is a ''symmetrization'' of not just one, but two, a priori unrelated, biholomorphic copies of \( \mathbb D^2 \). Explicit biholomorphisms between the bidisc and the top level domains of \( \mathcal D_1 \) are produced. A pleasant surprise was that the three dimensional hypersurfaces had been studied by Elie Cartan.

Date posted: January 2020.

The distinguished varieties and joint spectra of commuting matrices. This work is with Poornendu Kumar, a present student of mine and Haripada Sau, who finished his Ph.D. a few years ago with me. Dilation and von Neumann type inequalities are intimately intertwined. Following Ando's classical dilation theorem for a pair of commuting contractions, it is known that a von Neumann type inequality holds over the bidisc. Often, it is possible to use just a distinguished variety instead of the full bidisc in a von Neumann type inequality. We give a new description of distinguished varieties in the bidisc and use it for an improvement of the description of distinguished varieties in the symmetrized bidisc. We also characterize one dimensional distinguished varieties in the polydisc. The work depends on joint spectra of commuting matrices. This is especially gratifying since joint spectra played a major role in my thesis work more than two decades ago. It is remarkable that algebraic varieties which are basic objects in algebraic geometry can be so related with joint spectra of matrices, an essential ingredient of operator theory. At the root of our work is the Berger-Coburn-Lebow (BCL) theorem characterizing a commuting tuple of isometries. We use a lesser known realization formula for contractive holomorphic operator valued functions than the very famous one. It allows us to establish a certain one-to-one relationship between contractive holomorphic operator valued functions and model BCL triples.