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Arnab Patra

Publications and source records attributed to Arnab Patra.

17 recordsLinked to original sources
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 geometry of the algebraic Davis--Wielandt shell and norm-parallelism in $C^*$-algebra

This article is devoted to the study of the Davis--Wielandt shell and the Davis--Wielandt radii of elements in a $C^*$-algebra. Utilizing a state-space approach, several geometric properties of the algebraic Davis--Wielandt shell are established. Upper and lower bounds for the algebraic Davis--Wielandt radii are obtained including the Davis--Wielandt radius of the sum of $k$ elements. We also explore the relationship between norm-parallelism and the Davis--Wielandt radii of elements.

math.OA↗
arXiv2026

Convolution Operators on Weighted Hahn Spaces

This paper studies the convolution operator on weighted Hahn sequence spaces. The boundedness and compactness of these operators, together with the multiplier algebras of the weighted Hahn space and its dual, are investigated. A complete characterization of the spectrum and fine spectrum is obtained, with illustrative examples. The introduction of the weighted framework leads to the emergence of new multiplier and spectral properties.

math.FA↗
arXiv2026

$q$-Berezin Range of Operators in Hardy Space

This paper investigates the concept of the $q$-Berezin range and $q$-Berezin number of bounded linear operators acting on Hardy space. We obtain the $q$-Berezin range of some classes of operators on Hardy space. In addition, the convexity of the $q$-Berezin range is explored for finite-rank, diagonal, multiplication, weighted shift, and certain composition operators.

math.FA↗
arXiv2025

Spectral Properties of the Compact Rhaly and Compact Generalised Ces{\`a}ro Operators on Weighted $c_0$ Spaces

In this article, we conduct a comprehensive study on the continuity, compactness, and spectral properties of Rhaly operators and generalized Ces\`aro operators, acting on weighted null sequence spaces. We determine the point spectrum, continuous spectrum, and residual spectrum for compact Rhaly operators and compact generalized Ces\`aro operators. Additionally, we explore Goldberg's classifications of Rhaly operators over weighted null sequence spaces.

math.FA↗
arXiv2025

On the estimation of the $q$-numerical radius via Orlicz functions

This study utilizes Orlicz functions to provide refined lower and upper bounds on the q-numerical radius of an operator acting on a Hilbert space. Additionally, the concept of q-sectorial matrices is introduced and further bounds for the q-numerical radius are established. Our results unify several existing bounds for the q-numerical radius. Suitable examples are provided to supplement the estimations.

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

$q$-Numerical radius of sectorial matrices and $2 \times 2$ operator matrices

This article focuses on several significant bounds of $q$-numerical radius $w_q(A)$ for sectorial matrix $A$ which refine and generalize previously established bounds. One of the significant bounds we have derived is as follows: \[\frac{|q|^2\cos^2\alpha}{2} \|A^*A+AA^*\| \le w_q^2(A)\le \frac{\left(\sqrt{(1-|q|^2)\left(1+2sin^2(\alpha)\right)}+ |q|\right)^2}{2} \|A^*A+AA^*\|,\] where $ A $ is a sectorial matrix. Also, upper bounds for commutator and anti-commutator matrices and relations between $w_q(A^t)$ and $w_q^t(A)$ for non-integral power $t\in [0,1]$ are also obtained. Moreover, a few significant estimations of $q$-numerical radius of off-diagonal $2\times2$ operator matrices are developed.

math.FA↗
arXiv2024

Joint $q$-Numerical Ranges of Operators in Hilbert and Semi-Hilbert Spaces

This paper introduces and investigates the concept of the $q$-numerical range for tuples of bounded linear operators in Hilbert spaces. We establish various inequalities concerning the $q$-numerical radius associated with these operator tuples. Furthermore, we extend our study to define the $q$-numerical range in semi-Hilbert spaces and provide a proof of its convexity. Additionally, we explore several related results in this context.

math.FA↗
arXiv2023

New upper bounds for the $q$-numerical radius of Hilbert space operators

This article introduces several new upper bounds for the $q$-numerical radius of bounded linear operators on complex Hilbert spaces. Our results refine some of the existing upper bounds in this field. The $q$-numerical radius inequalities of products and commutators of operators follow as special cases. Finally, some new inequalities for the $q$-numerical radius of $2 \times 2$ operator matrices are established.

math.FA↗
arXiv2023

Spectral properties of the Rhaly operator on weighted null sequence spaces and associated operator ideals

In this article, a comprehensive study is made on the continuity, compactness, and spectrum of the lower triangular terraced matrix, introduced by H. C. Rhaly, Jr. [Houston J. Math. 15(1): 137-146, 1989], acting on the weighted null sequence spaces with bounded, strictly positive weights. Several spectral subdivisions such as point spectrum, residual spectrum, and continuous spectrum are also discussed. In addition, a new class of operator ideal $χ_{c_0(r)}^{(s)}$ associated to the Rhaly operator on weighted $c_0$ space is defined using the concept of $s$-number and it is proved that under certain condition, $χ_{c_0(r)}^{(s)}$ forms a quasi-Banach closed operator ideal.

math.SP↗
arXiv2023

Spectrum and Fine Spectrum of Band Matrices Generated by Oscillatory Sequences

In this paper, a new class of band matrices is considered where the entries of each non-zero band form a sequence with two limit points. The compact perturbation technique is used to study the spectrum over the $\ell_{p}, (1<p<\infty)$ sequence space. Several spectral subdivisions such as fine spectrum, discrete spectrum, essential spectrum, etc. are obtained. In addition, a few sufficient conditions on the absence of point spectrum over the essential spectrum are also discussed.

math.SP↗
arXiv2018

Spectrum and fine spectrum of generalised lower triangular triple band matrices over the sequence space $l_p$

The spectrum of triangular band matrices defined on the sequence spaces where the entries of each band is a constant or convergent sequence is well studied. In this article, the spectrum and fine spectrum of a new generalised difference operator defined by a lower triangular triple band matrix on the sequence space $l_p (1 \leq p < \infty)$ are obtained where the bands are considered as periodic sequences. The approximate point spectrum, defect spectrum, compression spectrum and the Goldberg classification of the spectrum are also discussed. Suitable examples are given in order to supplement the results. Several special cases of our findings are discussed which confirm that our study is more general and extensive.

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↗
arXiv2018

Spectra of the lower triangular matrix $\mathbb{B}(r_1,\dots , r_l; s_1, \dots, s_{l'})$ over $c_0$

The spectra and fine spectra of the lower triangular matrix $\mathbb{B}$ $(r_1,\dots , r_l;$ $ s_1, \dots, s_{l'})$ over the sequence space $c_0$ are determined. The diagonal and sub-diagonal entries of the matrix consist of two oscillatory sequences $r=(r_{k (\text{mod} \ l)+1})$ and $s= (s_{k(\text{mod} \ l')+1})$ respectively, whereas the rest of the entries of the matrix are zero. In particular, the spectra and fine spectra of the lower triangular matrix $\mathbb{B}(r_1,\dots , r_4; s_1, \dots, s_{6})$ over $c_0$ are discussed.

math.FA↗
arXiv2017

Relative Perturbation Bounds for the Joint Spectrum of Commuting Tuple of Matrices

In this paper, we study the relative perturbation bounds for joint eigenvalues of commuting tuples of normal $n \times n$ matrices. Some Hoffman-Wielandt type relative perturbation bounds are proved using the Clifford algebra technique. A result is also extended for diagonalizable matrices which improves a relative perturbation bound for single matrices.

math.FA↗