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.

**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.

**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 $\mathbb{D}^n$.

**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.

**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:

\begin{equation} k\cdot T:=\big(k_1(T_1, \ldots, T_d), \ldots, k_d(T_1, \ldots, T_d)\big),\,\, k\in K, \end{equation}

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.

**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.

**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.

Last updated: 08 Dec 2021