We cordially invite you to the symposium commemorating the superannuation of Professor Gadadhar Misra (IISc Bangalore). The (online) symposium is organised by Subrata Shyam Roy and Shibananda Biswas of IISER, Kolkata.

The programme schedule for the symposium is as follows:

Date: 30th July, 2021 (Friday)

Venue: Microsoft Teams (online)

Time Speaker & Title
10.00 am - 10.40 am Sameer Chavan     Positivity aspects of Dirichlet series
10.45 am - 11.25 am             Somnath Hazra     A family of homogeneous operators in the Cowen–Douglas class over the poly-disc
11.35 am - 12.15 pm Rajeev Gupta     The Caratheodory–Fejer interpolation on the polydisc
12.20 pm - 1.00 pm Surjit Kumar     $K$-homogeneous tuple of operators on bounded symmetric domains
1.00 pm - 2.15 pm Lunch
2.15 pm - 2.55 pm Ramiz Reza     Analytic $m$-isometries and weighted Dirichlet-type spaces
3.00 pm - 3.40 pm Jaydeb Sarkar     Analytic perturbations of unilateral shift

Each lecture will be of 40 minutes with 5 minutes break for Q&A and change of speaker.

### Abstracts

#### Lecture 1 ​

Speaker: Sameer Chavan (IIT Kanpur)

Title: ​ Positivity aspects of Dirichlet series

Abstract: In the first half of this talk, we discuss the space $\mathcal D[s]$ of finite Dirichlet series considered as a subspace of continuous functions on $\mathbb R_+$. Unlike the space of polynomials, $\mathcal D[s]$ fails to be an adapted space in the sense of Choquet. This causes an obstruction in identifying all positive linear functionals on $\mathcal D[s]$ as moment functionals (an analog of the so-called Riesz–Haviland Theorem). One solution (direct) to this problem can be based on a well-known one-point compactification technique in moment theory. Another solution (rather indirect) takes us to the topics like $\log$-moment sequences and Helson matrices of independent interest. In the second half, we focus on half-plane analog of the weighted Hardy spaces. The motivating example comes from the Riemann zeta function. We address the problem of finding members/multipliers of these spaces.

#### Lecture 2​

Speaker: Somnath Hazra (IISER Kolkata)

Title: ​ A family of homogeneous operators in the Cowen–Douglas class over the poly-disc

Abstract: ​ In this talk, we first describe a family of reproducing kernel Hilbert spaces of holomorphic functions taking values in $\mathbb{C}^r$ on the unit poly-disc $\mathbb{D}^n$ depending upon $r+n$ parameters of positive real numbers for any natural number $r$. It is then shown that these reproducing kernels are quasi-invariant with respect to the subgroup Möb$\times\cdots\times$Möb ($n$ times) of the bi-holomorphic automorphism group of $\mathbb{D}^n$. Using the quasi-invariant property, these reproducing kernels can be described explicitly. The adjoint of the $n$-tuples of multiplication operators by co-ordinate functions on these Hilbert spaces turn out to be homogeneous, irreducible, mutually unitarily inequivalent and in the Cowen-Douglas class over $\mathbb{D}^n$.

#### Lecture 3​​ ​

Speaker: Rajeev Gupta (IIT Goa)

Title: ​ The Caratheodory–Fejer interpolation on the polydisc

Abstract: CF-Problem: Given any polynomial $p$ in $n$-variables of degree $d$, find necessary and sufficient conditions on the coefficients of $p$ to ensure the existence of a holomorphic function $h$ defined on the polydisc such that $f:=p+h$ maps the polydisc into the unit disc in the complex plane and that for any multi-index $I$ with length at most $d$ $h^{(I)}(\boldsymbol 0)=0.$ In this talk, we give an algorithm for finding a solution to the Caratheodory–Fejer interpolation problem on the polydisc, whenever it exists. A necessary condition for the existence of a solution becomes apparent from this algorithm. Along the way a generalization of the well-known theorem due to Nehari will be obtained.

#### Lecture 4

Speaker: Surjit Kumar (IISc Bangalore)

Title: ​ $K$-homogeneous tuple of operators on bounded symmetric domains

Abstract: In this talk, we discuss Hilbert space operator tuples which are homogeneous under the action of a compact linear group.

Let $\Omega$ be an irreducible bounded symmetric domain of rank $r$ in $\mathbb C^d$ and $K$ is a maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\Omega$. The group $K$ consisting of linear transformations acts naturally on any $d$-tuple $T=(T_1, \ldots, T_d)$ of commuting bounded linear operators by the rule:

$$k\cdot T:=\big(k_1(T_1, \ldots, T_d), \ldots, k_d(T_1, \ldots, T_d)\big),\,\, k\in K,$$

where $k_1( z), \ldots, k_d( z)$ are linear polynomials.

If the orbit of this action modulo unitary equivalence is a singleton, then we say that $T$ is $K$-homogeneous. We obtain a model for a certain class of $K$-homogeneous $d$-tuple $T$ as the operators of multiplication by the coordinate functions $z_1,\ldots ,z_d$ on a $K$- invariant reproducing kernel Hilbert space of holomorphic functions defined on $\Omega$. Using this model we obtain a criterion for boundedness, unitary equivalence and similarity of these $d$-tuples.

This is joint work with Soumitra Ghara and Paramita Pramanick.

#### Lecture 5​​ ​

Speaker: Ramiz Reza (IISER Pune)

Title: ​ Analytic $m$-isometries and weighted Dirichlet-type spaces

Abstract: ​ We introduce a weighted Dirichlet-type space associated to any $(m − 1)$-tuple of finite, positive, Borel measures on the unit circle. We show that every cyclic, analytic $m$-isometry which satisfies a certain set of operator inequalities can be represented as an operator of multiplication by the coordinate function on such a weighted Dirichlet-type space. This extends a result of Richter on the class of cyclic analytic 2-isometries. Further we explore various properties of functions in these weighted Dirichlet type spaces.

#### Lecture 6​

Speaker: Jaydeb Sarkar (ISI Bangalore)

Title: ​ Analytic perturbations of unilateral shift

Abstract: The main aim of perturbation theory is to study (and also compare the properties of) $S:= F + T$, where $F$ is a finite rank (or compact, Hilbert-–Schmidt, Schatten–von Neumann class, etc.) operator and $T$ is a tractable operator (like unitary, normal, isometry, self-adjoint, etc.) on some Hilbert space. I will discuss joint work with Susmita Das in which we investigate some basic properties of shifts ($S$) that are finite rank ($F$) perturbations of the unilateral shift ($T$) on the classical Hardy space. Here shift ($S$) refers to the multiplication operator by the coordinate function $z$ on some analytic reproducing kernel Hilbert space defined on the open unit disc in the complex plane. Also, we will recall and introduce all the background material needed for this talk.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 22 Sep 2021