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V. M. Sholapurkar

Bio: V. M. Sholapurkar is an academic researcher from Sri Pratap College. The author has contributed to research in topics: Order (group theory) & Tuple. The author has an hindex of 3, co-authored 3 publications receiving 24 citations.

Papers
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TL;DR: In this paper, it is known that in dimension one, an invertible spherical hyperexpansion is unitary, and this rigidity theorem allows one to prove a variant of the Berger-Shaw Theorem which states that a finitely multi-cyclic spherical 2-isometry is essentially normal.
Abstract: The class of spherical hyperexpansions is a multi-variable analog of the class of hyperexpansive operators with spherical isometries and spherical 2-isometries being special subclasses. It is known that in dimension one, an invertible \(2\)-hyperexpansion is unitary. This rigidity theorem allows one to prove a variant of the Berger–Shaw Theorem which states that a finitely multi-cyclic \(2\)-hyperexpansion is essentially normal. In the present paper, we seek for multi-variable manifestations of this rigidity theorem. In particular, we provide several conditions on a spherical hyperexpansion which ensure it to be a spherical isometry. We further carry out the analysis of the rigidity theorems at the Calkin algebra level and obtain some conditions for essential normality of a spherical hyperexpansion. In the process, we construct several interesting examples of spherical hyperexpansions which are structurally different from the Drury-Arveson \(m\)-shift.

14 citations

Journal ArticleDOI
TL;DR: In this article, a class of completely hyperexpansive tuples of finite order is introduced, which is in some sense antithetical to the notion of completely hypercontractive tuplands of finite orders.

7 citations


Cited by
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TL;DR: For a bounded operator T on a Banach space X, the authors showed that β(m,p)(T,x)≤0 for all x∈X implies β (m−1,p,p)≥ 0 for all X ∈X.

32 citations

Journal ArticleDOI
TL;DR: In this article, the authors introduce two natural notions of Aluthge transforms (toral and spherical) for 2-variable weighted shifts and study their basic properties, and briefly discuss the relation between spherically quasinormal and isometric weighted shifts.

18 citations

Posted Content
TL;DR: In this paper, the multivariable counterpart of weighted shifts on rooted directed trees is developed. And the notion of spherically balanced multishifts on directed Cartesian product of rooted trees is introduced.
Abstract: We systematically develop the multivariable counterpart of the theory of weighted shifts on rooted directed trees. Capitalizing on the theory of product of directed graphs, we introduce and study the notion of multishifts on directed Cartesian product of rooted directed trees. This framework unifies the theory of weighted shifts on rooted directed trees and that of classical unilateral multishifts. Moreover, this setup brings into picture some new phenomena such as the appearance of system of linear equations in the eigenvalue problem for the adjoint of a multishift. In the first half of the paper, we focus our attention mostly on the multivariable spectral theory and function theory including finer analysis of various joint spectra and wandering subspace property for multishifts. In the second half, we separate out two special classes of multishifts, which we refer to as torally balanced and spherically balanced multishifts. The classification of these two classes is closely related to toral and spherical polar decompositions of multishifts. Furthermore, we exhibit a family of spherically balanced multishifts on $d$-fold directed Cartesian product $\mathscr T$ of rooted directed trees. These multishifts turn out be multiplication $d$-tuples $\mathscr M_{z, a}$ on certain reproducing kernel Hilbert spaces $\mathscr H_a$ of vector-valued holomorphic functions defined on the unit ball $\mathbb B^d$ in $\mathbb C^d$, which can be thought of as tree analogs of the multiplication $d$-tuples acting on the reproducing kernel Hilbert spaces associated with the kernels $\frac{1}{(1-\langle{z},{{w}\rangle})^a}~(z, w \in \mathbb B^d, a \in \mathbb N).$

16 citations

Journal ArticleDOI
TL;DR: In this paper, a class of operator tuples in complex Hilbert spaces, called spherical tuples, is introduced and studied, and a criterion for the Schatten Sp-class membership of cross-commutators of spherical m-shifts is defined.
Abstract: We introduce and study a class of operator tuples in complex Hilbert spaces, which we call spherical tuples. In particular, we characterize spherical multi-shifts, and more generally, multiplication tuples on RKHS. We further use these characterizations to describe various spectral parts including the Taylor spectrum. We also find a criterion for the Schatten Sp-class membership of cross-commutators of spherical m-shifts. We show, in particular, that cross-commutators of non-compact spherical m-shifts cannot belong to Sp for pm. We specialize our results to some well-studied classes of multi-shifts. We prove that the cross-commutators of a spherical joint m-shift, which is a q-isometry or a 2-expansion, belongs to Sp if and only if p > m. We further give an example of a spherical jointly hyponormal 2-shift, for which the cross-commutators are compact but not in Sp for any p < 1.

15 citations

Dissertation
16 Dec 2019
TL;DR: In this paper, the authors derived the positivity of the logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian, via results of quasi-factorization of the relative entropy.
Abstract: The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi- factorization results for the entropy. Inspired by the classical case, in this thesis we present a strategy to derive the positivity of the logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian, via results of quasi-factorization of the relative entropy. In particular, we address this problem for the heat-bath and Davies dynamics. In the first part of the thesis, we introduce the notion of conditional relative entropy in several ways, and subsequently use these concepts to obtain results of quasi- factorization of the relative entropy both in a weak and strong regime. Next, we show the positivity of logarithmic Sobolev constants for the heat-bath dynamics with tensor product fixed point, and then lift these results for the heat-bath dynamics in 1D and the Davies dynamics, showing that the first one is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi- factorization of the relative entropy, and the second one under some strong clustering of correlations. To conclude, in the last part of the thesis we study a notion related to the conditional relative entropy, namely the Belavkin-Staszewski relative entropy, for which we provide new conditions for equality in the data processing inequality, which we subsequently employ to strengthen the aforementioned inequality for the BS relative entropy in particular, and for maximal f-divergences in general. This thesis has been developped at Instituto de Ciencias Matematicas and Universidad Autonoma de Madrid under the supervision of David Perez-Garcia (U. Complutense de Madrid) and Angelo Lucia (Caltech).

14 citations