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Riddhick Birbonshi

Publications and source records attributed to Riddhick Birbonshi.

16 recordsLinked to original sources
arXiv2026

Numerical range and invariant subspaces of weighted composition operators on the Hardy space of Dirichlet series

In this paper, we study various results on numerical range of weighted composition operators on Hardy space of Dirichlet series. Certain conditions are provided under which the numerical range contains zero and some circular or elliptic disks. With the help of invariant subspaces of the operator, we also determine the numerical range of reductive weighted composition operators on this space. Additionally, we discuss some results on the Davis-Wielandt shell of composition operators with some natural questions that arise from our findings.

math.FA↗
arXiv2026

$q$-Berezin sectorial operators with applications to $q$-Berezin number inequalities and $q$-Berezin ranges

In this paper, we introduce a new class of operators, called $q$-Berezin sectorial operators, as an extension of the class of $q$-sectorial operators. By presenting examples on the Hardy-Hilbert space, we show that there exist operators which are $q$-Berezin sectorial but not $q$-sectorial. Several new inequalities for the $q$-Berezin number associated with this class of operators are also derived. In addition, we investigate the geometric structure of the $q$-Berezin range for various classes of operators on the Bergman space, including weighted shift and certain composition operators.

math.FA↗
arXiv2026

On the numerical radius of a class of weighted shift operators

In this paper, we derive some bounds on numerical radius of the weighted shift operator $T$ with weights $(1,sq,q^2,tq^3,q^4,sq^5,q^6,tq^7,\ldots)$ where $s,t >0$ and $0<q<1$. Furthermore, we provide an entire function $F_T(z)$. The reciprocal of the minimal positive root of $F_T(z)=0$ gives the numerical radius of $T$. These results generalize several previously known results on the numerical radius of weighted shift operators discussed in \cite{chakraborty2025numerical}.

math.FA↗
arXiv2026

Improved upper bounds for the Berezin numbers of operators on reproducing kernel Hilbert spaces

In this article, several upper bounds for the Berezin numbers of bounded linear operators on reproducing kernel Hilbert spaces are obtained through the use of interpolation paths of symmetric means and Orlicz functions. With suitable selections of these paths and functions, we show that the results presented here refine and generalize several earlier known findings. Furthermore, we derive some Berezin number inequalities for such operators using refined Young's inequalities.

math.FA↗
arXiv2026

Improved Bounds for numerical radius and $a$-numerical radius in ${C}^*$-algebras

In this article, we derive several significant upper bounds for the numerical radius and $a$-numerical radius of an element in a ${C}^*$-algebra by improving inequalities for positive linear functionals. Our findings refine and generalize the existing inequalities. Furthermore, we introduce a new notion to derive improved upper bounds of the numerical radius for an element in a ${C}^*$-algebra using the Moore-Penrose inverse.

math.FA↗
arXiv2026

On posinormality of weighted composition-differentiation operators on $H^2(\mathbb{D})$

In this article, the posinormality and coposinormality of weighted composition-differentiation operators on Hardy space $H^2(\mathbb{D})$ are investigated. It is observed that while a composition-differentiation operator $D_{ϕ,n}$ fails to be posinormal, the weighted composition-differentiation operator $D_{ψ,ϕ,n}$ can be posinormal for specific choices of $ψ, ϕ$. Some necessary conditions are obtained for posinormality and coposinormality of the operator $D_{ψ,ϕ,n}$. Furthermore, the adjoint formula for this operator is derived which also helped us to examine some results regarding posinormality of this operator.

math.FA↗
arXiv2026

Iterated Aluthge transforms of some composition operators on weighted Bergman spaces

In this paper, we compute the iterated Aluthge transforms $\widetilde{C_ϕ}^{(n)}$ of the composition operator $C_ϕ$ on the weighted Bergman spaces $\mathcal{A}_α^2(\mathbb{D})$, where $ϕ(z)=az+(1-a)$ for $0<a<1$. Also, we obtain the norm and numerical radius of $\widetilde{C_ϕ}^{(n)}$ on $\mathcal{A}_α^2(\mathbb{D})$. We establish that $\widetilde{C_ϕ}^{(n)}$ converges in the strong operator topology on $\mathcal{A}_α^2(\mathbb{D})$. The purpose of this paper is to examine the results of \cite{jung2015iterated} for the weighted Bergman spaces $\mathcal{A}_α^2(\mathbb{D})$. Additionally, by using the iterated Aluthge transforms of $C_ϕ^*$ on $\mathcal{A}_α^2(\mathbb{D})$, we derive the iterated Aluthge transforms of $C_σ$, where $\displaystyleσ(z)=\frac{az}{-(1-a)z+1}$ for $0<a<1$, on some weighted Hardy space $H^2(β_α)$ and study its convergence. Finally, we raise some questions that emerge from these findings.

math.FA↗
arXiv2026

An introduction of Berezin sectorial operators and its application to Berezin number inequalities

We introduce a new class of operators, called Berezin sectorial operators, which generalizes classical sectorial operators. We provide examples on the Hardy-Hilbert space showing that there exist operators that are Berezin sectorial but not sectorial and that the Berezin sectorial index can be strictly smaller than the classical one. We derive Berezin number inequalities for this class, including a weak version of the power inequality, and study geometric properties of the Berezin range for finite-rank and weighted shift operators on the Dirichlet space. We also raise the question of whether similar constructions are possible for composition-differentiation operators on the Dirichlet space.

math.FA↗
arXiv2026

A note on weighted composition operators on Dirichlet space

In this paper, we provide some sufficient conditions for the compactness of weighted composition operators on Dirichlet space. Furthermore, we characterize the numerical range of certain classes of weighted composition operators on Dirichlet space and establish the criteria that guarantee the inclusion of zero within the numerical range. Finally, several classes of weighted composition operators are introduced, whose numerical range contains either a circular disk with a given center and radius or an elliptical disk with specified foci and axis lengths.

math.FA↗
arXiv2025

On the $A$-$q$-Numerical Range of Operators in Semi-Hilbertian Spaces

This study investigates the $A$-$q$-numerical range of an operator within the framework of semi-Hilbertian spaces. Several fundamental properties of the $A$-$q$-numerical range are established, including spectral inclusion results and a disk union formula. Bounds for the $A$-$q$-numerical radius are derived, extending and generalizing previously known results. Finally, the notion of $A$-nilpotent operator is introduced, and it is shown that the $A$-$q$-numerical range of an $A$-nilpotent operator with index $2$ is a disk (open or closed) in the complex plane.

math.FA↗
arXiv2025

Numerical range of Toeplitz and Composition operators on weighted Bergman spaces

In this paper we completely describe the numerical range of Toeplitz operators on weighted Bergman spaces with harmonic symbol. We also characterize the numerical range of weighted composition operators on weighted Bergman spaces and classify some sets which are the numerical range of composition operators. We investigate the inclusion of zero in the numerical range, and compute the radius of circle and ellipse contained in the numerical range of weighted composition operators on weighted Bergman spaces.

math.FA↗
arXiv2024

A note on the $A$-numerical range of semi-Hilbertian operators

In this paper we explore the relation between the $A$-numerical range and the $A$-spectrum of $A$-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of $A$-normal operator and prove that closure of the $A$-numerical range of an $A$-normal operator is the convex hull of the $A$-spectrum. We further prove Anderson's theorem for the sum of $A$-normal and $A$-compact operators which improves and generalizes the existing result on Anderson's theorem for $A$-compact operators. Finally we introduce strongly $A$-numerically closed class of operators and along with other results prove that the class of $A$-normal operators is strongly $A$-numerically closed.

math.FA↗
arXiv2018

On some study of the fine spectra of generalized difference operator $Δ_{a,b}$ on $\ell_p \ (1<p<\infty)$

In this paper, we determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the generalized difference operator $Δ_{a,b}$ on the sequence space $\ell_p \ (1< p < \infty)$ where the real sequences $a=\{a_k\}$ and $b=\{b_k\}$ are not necessarily convergent. Hence our results generalize the work given by Akhmedov and El-Shabrawy [Math. Slovaca 65~(5) (2015) 1137--1152] for the sequence space $\ell_p (1< p <\infty)$.

math.FA↗
arXiv2017

A note on Anderson's theorem in the infinite-dimensional setting

Anderson's theorem states that if the numerical range W(A) of an n-by-n matrix A is contained in the unit disk and intersects with the unit circle at more than n points, then it coincides with the (closed) unit dissk. An analogue of this result for compact A in an infinite dimensional setting was established by Gau and Wu. We consider here the case of A being the sum of a normal and compact operator.

math.FA↗