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IIT Hyderabad

For a unitary matrix X of order n over the field of complex numbers and an entire function φ belonging to the Fock space F2: = F2(Cn) , we define an integral operator on F2(Cn) of the form (HφXf)(z)=∫Cnf(w)φ(z+X∗Xw¯)ezw¯dλ(w).Here dλ(z)=π-ne-|z|2dz is a Gaussian measure on Cn. We characterize all the symbols φ for which the operator HφX is bounded. Next, we consider integral operator on F2 defined by (Rφf)(z)=∫Cnf(w)φ(z⋆w¯)dλ(w)for φ∈ F2, where ⋆ is a coordinatewise multiplication. We give a complete characterization for the symbols φ∈ F2(Cn) so that the operator Rφ is bounded on F2. In addition to boundedness, we also obtain some fundamental results for the operators HφX and Rφ such as normality, the C∗-algebra properties, the spectrum and the compactness. Moreover, we characterize the common reducing subspaces for each of the collections BX={HφX∈B(F2):φ∈F2},R={Rφ∈B(F2):φ∈F2},respectively. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

World University root node

Department of Mathematics

Journal of Pseudo-Differential Operators and Applications

L -invariant and radial singular integral operators on the Fock space

Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if (Formula presented.). In this paper, first, we give a representation of C-normal operators on finite dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators and show that every compact C-normal operator has a solution for the eigenvalue problem. © 2022 Informa UK Limited, trading as Taylor &amp; Francis Group.

Linear and Multilinear Algebra

A representation of compact C-normal operators

We study the cyclic, supercyclic and hypercyclic properties of a composition operator Cφ on the Segal-Bargmann space H(ϵ), where φ(z) - Az + b, A is a bounded linear operator on ϵ, b ∈ ϵ with |A| ≤ 1 and A∗b belongs to the range of (I - A∗A)1/2. Specifically, under some conditions on the symbol φ we show that if Cφ is cyclic then A∗ is cyclic but the converse need not be true. We also show that if Cφ∗ is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space H(ϵ) for certain class of symbols φ. © 2022 G. Ramesh et al., published by De Gruyter.

Fulltext

Concrete Operators

Cyclic Composition operators on Segal-Bargmann space

In this note, we establish the cartesian decomposition of C-normal operator on a complex separable Hilbert space. Using this we give two characterizations of C-normal operators. These characterizations are useful in checking C-normality of an operator. We illustrate these results with examples. © 2021 Informa UK Limited, trading as Taylor &amp; Francis Group.

Cartesian decomposition of C-normal operators

We extend Hardy's theorem for rotation due to Lakey and Hogan for Dunkl transform associated with the reflection group (Formula presented.) in (Formula presented.) by using Dunkl-Hermite semigroup. In fact, we characterize (Formula presented.) by considering the decay of f and it's Dunkl transform (Formula presented.) along some lines in the complex plane, and we give a d-dimensional analogue for the same corresponding to the reflection group (Formula presented.) Moreover, we prove a version of Hardy's theorem, assuming the decay of Dunkl-Hermite coefficients. © 2020 Informa UK Limited, trading as Taylor & Francis Group.

Complex Variables and Elliptic Equations

Hardy's theorem and rotation for Dunkl transform

In the present article,we give an alternate and easier proof for the image characterization of L2(R2n) under the twisted Bargmann transform which was earlier studied by Krontz, Thangavelu and Xu. As a consequence, we study some properties of the twisted Bergman spaces for 0 < p ≤ ∞ and the Lp-boundedness of the twisted Bargmann transform, 1 ≤ p ≤ ∞. We also study Lp-boundedness of the twisted Bargmann projection Pt and the duality relations between the spaces Bpt(ℂ 2n), 1 < p < ∞. © 2021 Walter de Gruyter GmbH, Berlin/Boston

Forum Mathematicum

Study of twisted bargmann transform via bargmann transform

In this article, we consider the Schrödinger semigroup related to the Dunkl–Laplacian Δ μ (associated to finite reflection group G) on R n . We characterize the image of L2(Rn,eu2hμ(u)du) under the Schrödinger semigroup as a reproducing kernel Hilbert space. We define Dunkl–Sobolev space in L2(Rn,eu2hμ(u)du) and characterize it’s image under the Schrödinger semigroup associated to G=Z2n as a reproducing kernel Hilbert space up to equivalence of norms. Also we provide similar results for Schrödinger semigroup associated to Dunkl–Hermite operator. © 2017, Springer International Publishing AG, part of Springer Nature.

On the images of Dunkl–Sobolev spaces under the Schrödinger semigroup associated to Dunkl operators

We give the necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss the structure of such operators in the case of self-adjoint and normal operators separately. Finally, we discuss several properties of absolutely norm attaining operators. © 2019, Indian Academy of Sciences.

Proceedings of the Indian Academy of Sciences: Mathematical Sciences

On absolutely norm attaining operators

In this article, we consider the Schrödinger semigroup for the Laplacian Δ on Rn, and characterize the image of a Sobolev space in L2(Rn,eu2du) under this semigroup as weighted Bergman space (up to equivalence of norms). Also we have a similar characterization for Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator H on Rn. © 2018 Walter de Gruyter GmbH, Berlin/Boston.

Advances in Pure and Applied Mathematics

On the images of Sobolev spaces under the Schrödinger semigroup

In this article, we discuss Benedicks' theorem for the Weyl transform associated with the Heisenberg group. Suppose σ ∈ L2 (Hn × R, S2, dν × dμ) be a function with compact support and if the Weyl transform Wσ is of finite rank, then we show that σ is zero almost everywhere on Hn × R. © 2018 Informa UK Limited, trading as Taylor & Francis Group.

Integral Transforms and Special Functions

Benedicks' theorem for the Weyl Transform associated with the Heisenberg group

This article deals with Lp-L2 estimates for solutions of the wave equation associated to the Grushin operator G := [EQUATION PRESENTED]. We make use of the Lp boundedness of Riesz transforms for the Grushin operator in studying the wave equation. © 2018 Walter de Gruyter GmbH, Berlin/Boston.

Lp-L 2 estimates for solutions of the wave equation associated to the Grushin operator

The image of a subspace of L2(Rn+1) under Grushin semigroup is characterized as direct sum of two weighted Bergman spaces. Hermite Sobolev space of positive order on Rn is defined, and the image of this space under Hermite semigroup is found out using Caputo fractional derivative as weighted Bergman space. The connection of Grushin operator and parametrized Hermite operator is used in this paper extensively and with that help image of Grushin Sobolev space of positive order is characterized. © 2017, Springer International Publishing.

Images of some subspaces of L2(Rm) under Grushin and Hermite semigroup

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$\end{document} under the heat kernel transform on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n,$\end{document} using direct sum and direct integral of Bergmann spaces and certain unitary representations of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} which can be realized on the Hilbert space of Hilbert-Schmidt operators on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2(\mathbb {R}^n).$\end{document} We also show that the image of Sobolev space of negative order \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$\end{document} is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} under the heat kernel transform. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Mathematische Nachrichten

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

This paper deals with the Schrödinger equation i∂su(z, t; s) - Lu(z, t; s)= 0, where L is the sub-Laplacian on the Heisenberg group. Assume that the initial data f satisfies |f(z, t)| ≤ qa(z, t), where qs is the heat kernel associated to L. If in addition |u(z, t; s0) ≤ qβ (z, t), for some s0 ∈ R \ {0}, then we prove that u(z, t; s)= 0 for all s ∈ R whenever αβ &lt; s02. This result holds true in the more general context of H-type groups. We also prove an analogous result for the Grushin operator on Rn+1. © 2013 Australian Mathematical Publishing Association Inc.

Journal of the Australian Mathematical Society

Uniqueness of solutions to schrödingera equations on H-type groups

Let A denote a real linear transformation on Cn which is symmetric and positive-definite relative to the real inner product Re〈z,w〉, z,w∈Cn. Let FA(Cn) denote the Fock space consisting of holomorphic functions on Cn which are square integrable with respect to the Gaussian measure dμA(z)=detRAΠne-Re〈Az,z〉. For w∈Cn, let ewA(z)=eA(z,w)=e-12Re〈Aw,w〉KA(z,w), z∈Cn, where KA is the reproducing kernel for FA(Cn). The main aim of this paper is to show that there exist a,b>0 such that the set of functions {ema+inbA:m,n∈Zn} forms a frame in FA. © 2010 Elsevier Inc.

Journal of Mathematical Analysis and Applications

Frames in generalized Fock spaces

Let FA(Cn) denote the Fock space associated with a real linear transformation A on Cn which is symmetric and positive definite relative to the real inner product Re〈z,w〉, z,w∈Cn. Let BA denote the Bargmann transform, mapping L2(Rn) unitarily onto FA(Cn). In this note, we show that one can find a group G, whose unitary irreducible representation at its base vector coincides with BA*Kw up to a constant multiple, where BA* denotes the adjoint of BA and Kw denotes the reproducing kernel of FA(Cn). © 2010 Elsevier Masson SAS.

Bulletin des Sciences Mathematiques

Generalized Bargmann transform and a group representation

In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space Wk,p(Hn) coincides with the class of right Fourier multipliers for Lp(Hn) for k ∈ ℕ, 1 &lt; p &lt; ∞. Towards this end, it is shown that the operators RjR̄jℒ-1 and RjR̄jRjℒ-1 are bounded on Lp(Hn), 1 &lt; p &lt; ∞, ℒ is the sublaplacian on Hn. This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space Wk,1(Hn) coincides with the dual space of the projective tensor product of two function spaces. © 2010 Akadémiai Kiadó, Budapest, Hungary.

Analysis Mathematica

Fourier multipliers for Sobolev spaces on the Heisenberg group

It is shown that the Hermite 'polynomial' semigroup {e-tℍ: t &gt; 0} maps Lp(ℝn, ρ into the space of holomorphic functions in Lr(ℂn, Vt,p/2(r+e)/2) for each ε &gt; 0, where ρ is the Gaussian measure, Vt,p/2r+e/2 is a scaled version of Gaussian measure with r = p if 1 &lt; p &lt; 2 and r = p′ if 2 &lt; p &lt; ∞ with 1/p + 1/p′ = 1. Conversely if F is a holomorphic function which is in a "slightly" smaller space, namely Lrℂn, Vt,p/2r/2), then it is shown that there is a function f ∈ Lpℝn, ρ) such that e-tH f = F. However, a single necessary and sufficient condition is obtained for the image of L2(ℝn,Pp/2 under e-tH, 1 &lt; p &lt; ∞. Further it is shown that if F is a holomorphic function such that F ε L1 (ℂn, Vt,p/21/2 or F ε Lm1,p ℝ2n, then there exists a function f ε Lp (ℝn,p) such that e-tℍf = F, where m(x,y), e-x2/ (p-1)e4t+1e-y2/e4t-1 and 1 &lt; p &lt; ∞.