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E. K. Narayanan

Publications and source records attributed to E. K. Narayanan.

18 recordsLinked to original sources
arXiv2025

Zero Products of Toeplitz operators on the Hardy and Bergman spaces over an annulus

We study the zero product problem of Toeplitz operators on the Hardy space and Bergman space over an annulus. Assuming a condition on the Fourier expansion of the symbols, we show that there are no zero divisors in the class of Toeplitz operators on the Hardy space of the annulus. Using the reduction theorem due to Abrahamse, we characterize compact Hankel operators on the Hardy space of the annulus, which also leads to a zero product result. Similar results are proved for the Bergman space over the annulus.

math.FA↗
arXiv2024

Contractive Hilbert modules on quotient domains

Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $\mathbb D^n$ in $\mathbb C^n.$ A $\boldsymbolΘ_n$-contraction is a commuting tuple of operators on a Hilbert space having $$\overline{\boldsymbolΘ}_n:=\{\boldsymbolθ(z)=(θ_1(z),\ldots,θ_n(z)):z\in\overline{\mathbb D}^n\}$$ as a spectral set, where $\{θ_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G(m,p,n).$ A plethora of examples of $\boldsymbolΘ_n$-contractions is exhibited. Under a mild hypothesis, it is shown that these $\boldsymbolΘ_n$-contractions are mutually unitarily inequivalent. These inequivalence results are obtained concretely for the weighted Bergman modules under the action of the permutation groups and the dihedral groups. The division problem is shown to have negative answers for the Hardy module and the Bergman module on the bidisc. A Beurling-Lax-Halmos type representation for the invariant subspaces of $\boldsymbolΘ_n$-isometries is obtained.

math.FA↗
arXiv2024

Mackey Imprimitivity and commuting tuples of homogeneous normal operators

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting $d$- tuples of homogeneous normal operators. The Hahn-Hellinger theorem gives a canonical decomposition of a $*$- algebra representation $ρ$ of $C_0(\mathbb{S})$ (where $\mathbb S$ is a locally compact Hausdorff space) into a direct sum. If there is a group $G$ acting transitively on $\mathbb{S}$ and is adapted to the $*$- representation $ρ$ via a unitary representation $U$ of the group $G$, in other words, if there is an imprimitivity, then the Hahn-Hellinger decomposition reduces to just one component, and the group representation $U$ becomes an induced representation, which is Mackey's imprimitivity theorem. We consider the case where a compact topological space $S\subset \mathbb {C}^d$ decomposes into finitely many $G$- orbits. In such cases, the imprimitivity based on $S$ admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of $G$- orbits.

math.FA↗
arXiv2024

Unitary parts of Toeplitz operators with operator-valued symbols

Motivated by the canonical decomposition of contractions on Hilbert spaces, we investigate when contractive Toeplitz operators on vector-valued Hardy spaces on the unit disc admit a non-zero reducing subspace on which its restriction is unitary. We show that for a Hilbert space $\mathcal{E}$ and operator-valued symbol $Φ\in L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T})$, the Toeplitz operator $T_Φ$ on $H_{\mathcal{E}}^2(\mathbb{D})$ has such a unitary subspace if and only if there exists a Hilbert space $\mathcal{F}$, an inner function $Θ(z) \in H_{\mathcal{B}(\mathcal{F}, \mathcal{E})}^{\infty}(\mathbb{D})$, and a unitary $U:\mathcal{F} \rightarrow \mathcal{F}$ such that \[ Φ(e^{it}) Θ(e^{it}) = Θ(e^{it}) U \quad \text{and} \quad Φ(e^{it})^* Θ(e^{it}) = Θ(e^{it}) U^* \quad (\text{ a.e. on }\mathbb{T}). \] This result can be seen as a generalization of the corresponding result for Toeplitz operators on $H^2(\mathbb{D})$ by Goor in [13]. We provide finer characterizations for analytic Toeplitz operators by finding the correspondence between the unitary parts of $T_Φ$ on $H_{\mathcal{E}}^2(\mathbb{D})$ and $Φ(0)$ on $\mathcal{E}$.

math.FA↗
arXiv2023

Toeplitz operators on the weighted Bergman spaces of quotient domains

Let $G$ be a finite pseudoreflection group and $Ω\subseteq \mathbb C^d$ be a bounded domain which is a $G$-space. We establish identities involving Toeplitz operators on the weighted Bergman spaces of $Ω$ and $Ω/G$ using invariant theory and representation theory of $G.$ This, in turn, provides techniques to study algebraic properties of Toeplitz operators on the weighted Bergman space on $Ω/G.$ We specialize on the generalized zero-product problem and characterization of commuting pairs of Toeplitz operators. As a consequence, more intricate results on Toeplitz operators on the weighted Bergman spaces on some specific quotient domains (namely symmetrized polydisc, monomial polyhedron, Rudin's domain) have been obtained.

math.CV↗
arXiv2022

Projective representations of Heisenberg groups over the rings of order p^2

In this article we describe the 2-cocycles, Schur multiplier and representation group of discrete Heisenberg groups over the unital rings of order $p^2$. We describe all projective representations of Heisenberg groups with entries from the rings $\mathbb Z/p^2\mathbb Z$ and $\mathbb{F}_p[t]/(t^2)$ and obtain a classification of their degenerate and non-degenerate 2-cocycles.

math.GR↗
arXiv2021

Injectivity of spherical means on H-type groups

We establish injectivity results for three different spherical means on an $H$-type group, $G$. First is the standard spherical means which is defined to be the average of a function over the spheres in the complement of the center, second is the average over the product of spheres in the center and its complement, and the third is the average over the spheres defined by a homogeneous norm on $G$. If $m$ is the dimension of the center of $G$, injectivity of these spherical means is proved for the range $1 \leq p \leq \frac{2m}{m-1}$. Examples are provided to show the sharpness of our results in the first two cases.

math.FA↗
arXiv2021

Hypergeometric functions of type $BC$ and standard multiplicities

We study the Heckman-Opdam hypergeometric functions associated to a root system of type $BC$ and a multiplicity function which is allowed to assume some non-positive values (a standard multiplicity function). For such functions, we obtain positivity properties and sharp estimates which imply a characterization of the bounded hypergeometric functions. As an application, our results extend known properties of Harish-Chandra's spherical functions on Riemannian symmetric spaces of the non-compact type $G/K$ to spherical functions over homogeneous vector bundles on $G/K$ which are associated to certain small $K-$types.

math.RT↗
arXiv2017

Differential operators, radial parts and a one-parameter family of hypergeometric functions of type BC

We introduce the symmetric (respectively, non-symmetric) $τ_{-\ell}-$hypergeometric functions associated with a root system of type $BC$ as joint eigenfunctions of a commutative algebra of differential (respectively, differential-reflection) operators. Under certain conditions on the real parameter $\ell$, we derive their properties (positivity, estimates, asymptotics and boundedness) by establishing the analogous properties for the Heckman-Opdam (symmetric and non-symmetric) hypergeometric functions corresponding to (not necessarily positive) multiplicity functions which are standard.

math.RT↗
arXiv2016

On monomial representations of finitely generated nilpotent groups

A result of D. Segal states that every complex irreducible representation of a finitely generated nilpotent group $G$ is monomial if and only if $G$ is abelian-by-finite. A conjecture of A. N. Parshin, recently proved affirmatively by I.V. Beloshapka and S. O. Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture combining the ideas of I. D. Brown and P. C. Kutzko. We also characterize finite dimensional irreducible representations of two step nilpotent groups and also provide a full description of the finite dimensional representations of two step groups whose center has rank one.

math.RT↗
arXiv2016

On characterization of monomial representations of discrete supersolvable groups

We prove that an abstract (possibly infinite dimensional) complex irreducible representation of a discrete supersolvable group is monomial if and only if it has finite weight. We also prove a general result that implies converse of Schur's lemma holds true for certain induced representations of finitely generated discrete groups. At last, we work out example of infinite dihedral group and prove that it is a monomial group.

math.RT↗
arXiv2016

Analytic Model of Doubly Commuting Contractions

An n-tuple (n \geq 2), T = (T_1, \ldots, T_n), of commuting bounded linear operators on a Hilbert space \mathcal{H} is doubly commuting if T_i T_j^* = T_j^* T_i for all $1 \leq i < j \leq n$. If in addition, each T_i \in C_{\cdot 0}, then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple $T$ can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space H^2_{\mathcal{D}_{T^*}}(\mathbb{D}^n). As a consequence of the dilation theorem, we prove that there exists a closed subspace \mathcal{S}_T of the form \[\mathcal{H}_{T} := \sum_{i=1}^n Φ_{T_i} H^2_{\mathcal{E}_{T_i}}(\mathbb{D}^n),\] where \{\mathcal{E}_{T_i}\}_{i=1}^n are Hilbert spaces, Φ_{T_i} \in H^\infty_{\mathcal{B}(\mathcal{E}_{T_i}, \mathcal{D}_{T^*})}(\mathbb{D}^n) such that each Φ_{T_i} (1 \leq i \leq n) is either a one variable inner function in z_i, or the zero function. Moreover, \mathcal{H} \cong \mathcal{S}_T^\perp and \[(T_1, \ldots, T_n) \cong P_{\mathcal{S}_T^\perp} (M_{z_1}, \ldots, M_{z_n})|_{\mathcal{S}_T^\perp}.\]

math.FA↗
arXiv2013

Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications

A series expansion for Heckman-Opdam hypergeometric functions $φ_λ$ is obtained for all $λ\in \mathfrak a^*_{\mathbb C}.$ As a consequence, estimates for $φ_λ$ away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The $L^p$-theory for the hypergeometric Fourier transform is developed for $0<p<2$. In particular, an inversion formula is proved when $1\leq p <2$.

math.RT↗
arXiv2010

Support theorem on R^n and non compact symmetric spaces

We consider convolution equations of the type f * T = g where f, g are in L^p(R^n) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T we show that f is compactly supported, provided g is. Similar results are proved for non compact symmetric spaces as well.

math.FA↗
arXiv2009

Benedick's theorem for the Heisenberg group

If $f$ is a compactly supported function on the Heisenberg group and the group Fourier transform $\hat{f}(λ)$ is a finite rank operator for all $λ$ then $f$ is the zero function.

math.FA↗
arXiv2009

Segal-Bargmann transform and Paley-Wiener theorems on $M(2).$

We study the Segal-Bargmann transform on $M(2).$ The range of this transform is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are studied. Using a Gutzmer type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification of $M(2).$ We also prove a Paley-Wiener theorem for the inverse Fourier transform

math.FA↗