arXiv subjects

V. B. Kiran Kumar

Publications and source records attributed to V. B. Kiran Kumar.

11 recordsLinked to original sources

arXiv2026

Korovkin-type approximation for non-positive operators

The classical Korovkin theorem traditionally relies on the positivity of the underlying sequence of operators. In 1968, D. E. Wulbert obtained a non-positive version by exploiting geometric properties of function spaces, namely the Choquet boundary and the unique extension property of extreme points of the dual unit ball for weakly separating subspaces. In this article we develop this geometric approach further and prove a Korovkin-type theorem for uniformly bounded sequences of operators on $C(X)$ and on $L^1[0,1]$, with the convergence of the operators on a test set being replaced by convergence to a limit operator. The main emphasis is on the underlying geometry: the weakly separating subspace, its Choquet boundary, and the associated unique extension property. As an application, we show that the Grünwald interpolation operators, which are non-positive, satisfy a Korovkin-type approximation theorem. We also extend this to an $L^1(\mathbb{R})$ setting and include numerical illustrations.

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arXiv2024

Revisiting Korovkin-type Theorems in Banach Function Spaces

This article delves into Korovkin-type theorems in Banach function spaces, as established by Yusuf Zeren et al. (2022). We prove that in this theorem, the positivity of the operators is not a necessary requirement and provide example of a non positive operator where it is applicable. Under the assumption of positivity, we establish an operator version of the result. Additionally, we derive a quantitative form of the result using the modulus of continuity. We apply the result to examples such as Lebesgue space, Weighted Lebesgue space, Grand Lebesgue space, etc. Furthermore, we present numerical illustrations for specific cases.

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arXiv2024

A Note on Generalized Locally Toeplitz Operators

Generalized Locally Toeplitz (GLT) matrix sequences arise from large linear systems that approximate Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), and Integro-Differential Equations (IDEs). GLT sequences of matrices have been developed to study the spectral/singular value behaviour of the numerical approximations to various PDEs, Fades and IDEs. These approximations can be achieved using any discretization method on appropriate grids through local techniques such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, and Discontinuous Galerkin methods. Spectral and singular value symbols are essential for analyzing the eigenvalue and singular value distributions of matrix sequences in the Weyl sense. In this article, we provide a comprehensive overview of the operator-theoretic aspect of GLT sequences. The theory of GLT sequences, along with findings on the asymptotic spectral distribution of perturbed matrix sequences, is a highly effective and successful method for calculating the spectral symbol f. Therefore, developing an automatic procedure to compute the spectral symbols of these matrix sequences would be advantageous, a task that Ahmed Ratnani, N S Sarathkumar, S. Serra-Capizzano have partially undertaken. As an application of the theory developed here, we propose an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming they form a GLT sequence that meets mild conditions.

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arXiv2024

An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators

In this article, we obtain some results in the direction of ``infinite dimensional symplectic spectral theory". We prove an inequality between the eigenvalues and symplectic eigenvalues of a special class of infinite dimensional operators. Let $T$ be an operator such that $T - αI$ is compact for some $α> 0$. Denote by $\{{λ_j^R}^\downarrow(T)\}$, the set of eigenvalues of $T$ lying strictly to the right side of $α$ arranged in the decreasing order and let $\{{λ_j^L}^\uparrow(T)\}$ denote the set of eigenvalues of $T$ lying strictly to the left side of $α$ arranged in the increasing order. Furthermore, let $\{{d_j^R}^\downarrow(T)\}$ denote the symplectic eigenvalues of $T$ lying strictly to the right of $α$ arranged in decreasing order and $\{{d_j^L}^\uparrow(T)\}$ denote the set of symplectic eigenvalues of $T$ lying strictly to the left of $α$ arranged in increasing order, respectively (such an arrangement is possible as it will be shown that the only possible accumulation point for the symplectic eigenvalues is $α$). Then we show that $${d_j^R}^\downarrow(T) \leq {λ_j^R}^\downarrow(T), \quad j = 1,2, \cdots, s_r$$ and $${λ_j^L}^\uparrow(T) \leq {d_j^L}^\uparrow(T), \quad j = 1,2, \cdots, s_l,$$ where $s_r$ and $s_l$ denote the number of symplectic eigenvalues of $T$ strictly to the right and left of $α$, respectively. This generalizes a finite dimensional result obtained by Bhatia and Jain (J. Math. Phys. 56, 112201 (2015)). The class of Gaussian Covariance Operators (GCO) and positive Absolutely Norm attaining Operators ($(\mathcal{A}\mathcal{N})_+$ operators) appear as special cases of the set of operators we consider.

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arXiv2024

Stability in Non-Normal Periodic Jacobi Operators: Advancing Börg's Theorem

Periodic Jacobi operators naturally arise in numerous applications, forming a cornerstone in various fields. The spectral theory associated with these operators boasts an extensive body of literature. Considered as discretized counterparts of Schrödinger operators, widely employed in quantum mechanics, Jacobi operators play a crucial role in mathematical formulations. The classical uniqueness result by G. Börg in $1946$ occupies a significant place in the literature of inverse spectral theory and its applications. This result is closely intertwined with M. Kac's renowned article, 'Can one hear the shape of a drum?' published in $1966$. Since $1975,$ discrete versions of Börg's theorem have been available in the literature. In this article, we concentrate on the non-normal periodic Jacobi operator and the discrete versions of Börg's Theorem. We extend recently obtained stability results to encompass non-normal cases. The existing stability findings establish a correlation between the oscillations of the matrix entries and the size of the spectral gap. Our result encompasses the current self-adjoint versions of Börg's theorem, including recent quantitative variations. Here, the oscillations of the matrix entries are linked to the path-connectedness of the pseudospectrum. Additionally, we explore finite difference approximations of various linear differential equations as specific applications.

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arXiv2023

A Note on Approximating the Symplectic Spectrum

The symplectic eigenvalues play a significant role in finite mode quantum information theory, and Williamson normal form proves to be a valuable tool in this area. Understanding the symplectic spectrum of a Gaussian Covariance Operator is a crucial task. Recently, an infinite-dimensional analogue of Williamson Normal form was discovered, which has been instrumental in studying infinite mode Gaussian quantum states. However, most existing results pertain to finite-dimensional operators, leaving a dearth of literature in the infinite-dimensional context. The focus of this article is on employing approximation techniques to estimate the symplectic spectrum of certain infinite-dimensional operators. These techniques are well-suited for a particular class of operators, including specific types of infinite mode Gaussian Covariance Operators. Our approach involves computing the Williamson Normal form and deriving bounds for the symplectic spectrum of these operators. As a practical application, we explicitly compute the symplectic spectrum of Gaussian Covariance Operators. Through this research, we aim to contribute to the understanding of symplectic eigenvalues in the context of infinite-dimensional operators, opening new avenues for exploration in quantum information theory and related fields.

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arXiv2022

Operator version of Korovkin Theorem; Degree of Convergence and its Applications

In a recent article, Dumitru Popa proved an operator version of the Korovkin theorem. We recall the quantitative version of the Korovkin theorem obtained by O. Shisha and B. Mond in 1968. In this paper, we obtain a quantitative estimate for the operator version of the Korovkin theorem obtained by Dumitru Popa. We also consider various examples where the operator version is applicable and obtain similar estimates leading to the degree of convergence. In addition, we obtain the trigonometric analogue of this result by proving the quantitative version. Finally, we apply this result to the preconditioning problem of large linear systems with the Toeplitz structure.

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arXiv2021

Banach Spaces of GLT Sequences and Function Spaces

The Generalized Locally Toeplitz (GLT) sequences of matrices have been originated from the study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate some partial differential equations by discretization. The study of the asymptotic spectral behaviour of GLT sequence is very important in analysing the solution of corresponding partial differential equations. The approximating classes of sequences (a.c.s) and the spectral symbols are important notions in this connection. Recently, G. Barbarino obtained some additional results regarding the theoretical aspects of such notions. He obtained the completeness of the space of matrix sequences with respect to pseudo metric a.c.s. Also, he identified the space of GLT sequences with the space of measurable functions. In this article, we follow the same research line and obtain various results connecting the sub-algebras of matrix sequence spaces and sub-algebras of function spaces. In some cases, these are identifications as Banach spaces and some of them are Banach algebra identifications. In the process, we also prove that the convergence notions in the sense of eigenvalue/singular value clustering are equivalent to the convergence with respect to the metrics introduced here. These convergence notions are related to the study of preconditioners in the case of matrix/operator sequences. Finally, as an application of our main results, we establish a Korovkin-type result in the setting of GLT sequences.

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arXiv2019

A note on discrete Borg-type theorems

We consider the discrete versions of the well known Borg theorem and use simple linear algebraic techniques to obtain new versions of the discrete Borg type theorems. To be precise, we prove that the periodic potential of a discrete Schrodinger operator is almost a constant if and only if the possible spectral gaps of the operator are of small width. This result is further extended to more general settings and the connection to the well known Ten Martini problem is also discussed.

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arXiv2017

Approximation of Entropy Numbers

The purpose of this article is to develop a technique to estimate certain bounds for entropy numbers of diagonal operator on spaces of p-summable sequences for finite p greater than 1. The approximation method we develop in this direction works for a very general class of operators between Banach spaces, in particular reflexive spaces. As a consequence of this technique we also obtain that the entropy number of a bounded linear operator T between two separable Hilbert spaces is equal to the entropy number of the adjoint of T. This gives a complete answer to the question posed by B. Carl [4] in the setting of separable Hilbert spaces.

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arXiv2012

Perturbation of operators and approximation of spectrum

Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)_n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered here is to approximate the spectrum of A(x) using the sequence of eigenvalues of A(x)_n. We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of A(x) can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of A(x)_n. The known results for a bounded selfadjoint operator, are translated into the case of a holomorphic family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of A(0) = A and study the effect of holomorphic perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz-Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here.

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