arXiv subjects

M. N. N. Namboodiri

Publications and source records attributed to M. N. N. Namboodiri.

7 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.

math.FA↗

arXiv2025

Concrete Quantum Channels and Algebraic Structure of Abstract Quantum Channels

This article analyzes the algebraic structure of the set of all quantum channels and its subset consisting of quantum channels that have Holevo representation. The regularity of these semigroups under composition of mappings is analyzed. It is also known that these sets are compact convex sets and, therefore, rich in geometry. An attempt is made to identify generalized invertible channels and also the idempotent channels. When channels are of the Holevo type, these two problems are fully studied in this article. The motivation behind this study is its applicability to the reversibility of channel transformations and recent developments in resource-destroying channels, which are idempotents. This is related to the coding-encoding problem in quantum information theory. Several examples are provided, with the main examples coming from pre-conditioner maps which assign preconditioners to matrices in numerical linear algebra. Thus, the known pre-conditioner maps are viewed as quantum channels in finite dimensions. In addition, the infinite-dimensional analogue of preconditioners is introduced and certain limit theorems are discussed; this is with an aim to analyze asymptotic methods in quantum channels analogous to problems in asymptotic linear algebra.

quant-ph↗

arXiv2022

Regularity Of The Semi-group Of Regular Probability Measures On Compact Hausdorff Topological Groups

There are many deep results on the structure of REGULAR probability measures $P(G)$ on compact/locally compact, Hausdorff topological groups G. See, for instance, the classic monographs by KR Parthasarathy, Ulf Grenander, A.Mukherjea and Nicolas A.Tserpes. It is known that the set $P(G)$ forms a semi-group under convolution. Wendel in his remarkable paper, proved a basic result regarding support of convolution of two probability measures. Consequently, he established that the semi-group $P(G)$ is not a group. In this short paper, it is proved that for a compact topological group G, the semi-group P(G) of probability measures is not algebraically regular. However, there are concrete regular semi-groups in which P(G) can be embedded.

math.FA↗

arXiv2018

On continuous movement of the discrete spectrum of Schrödinger operators

Continuous movement of discrete spectrum of the Schrödinger operator $H(z)=-\frac{d^2} {dx^2}+V_0+z V_1$, with $\int_0^\infty {x |V_j(x)| dx} < \infty$, on the half-line is studied as $z$ moves along a continuous path in the complex plane. The analysis provides information regarding the members of the discrete spectrum of the non-selfadjoint operator that are evolved from the discrete spectrum of the corresponding selfadjoint operator.

math.SP↗

arXiv2016

Quasi Hyperrigidity and Weak Peak Points for Non-Commutative Operator Systems

In this article, we introduce the notions of weak boundary repre- sentation, quasi hyperrigidity and weak peak points in the non-commutative setting for operator systems in C* algebras. An analogue of Saskin theorem relating quasi hyperrigidity and weak Choquet boundary for particular classes of C* algebras is proved. We also show that, if an irreducible representation is a weak boundary representation and weak peak then it is a boundary repre- sentation. Several examples are provided to illustrate these notions. It is also observed that isometries on Hilbert spaces play an important role in the study of certain operator systems.

math.OA↗

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.

math.FA↗

arXiv2012

Korovkin results and Frobenius optimal approximants for infinite dimensional bounded linear operators

The classical as well as non commutative Korovkin-type theorems deal with convergence of positive linear maps with respect to modes of convergences such as norm convergence and weak operator convergence. In this article, Korovkin-type theorems are proved for convergence of completely positive maps with respect to weak, strong and uniform clustering of sequences of matrices of growing order. Such modes of convergence were originally considered for Toeplitz matrices (see [23],[26]). As an application, we translate the Korovkin-type approach used in the finite dimensional case, in the setting of preconditioning large linear systems with Toeplitz structure, into the infinite dimensional context of operators acting on separable Hilbert spaces. The asymptotic of these pre-conditioners are obtained and analyzed using the concept of completely positive maps. It is observed that any two limit points of the same sequence of pre-conditioners are the same modulo compact operators. Finally, we prove the generalized versions of the Korovkin type theorems in [23]. Keywords: Completely positive maps, Frobenius norm, Pre-conditioners.

math.FA↗