Refinements and reverses of Hölder-McCarthy operator inequality via a Cartwright-field result

By the use of a classical result of Cartwright and Field we obtain in this paper some new refinements and reverses of Hölder-McCarthy operator inequality in the case p 2 (0; 1). A comparison for the two upper bounds obtained showing that neither of them is better in general, is also performed

Operator inequalities, functional models and ergodicity

We discuss when an operator T, subject to a rather general inequality in hereditary form, admits a unitarily equivalent functional model of Agler type in the reproducing kernel Hilbert space associated to the inequality. The kernel need not be of Nevanlinna-Pick type. We define a defect operator D in our context and discuss the structure of the spectrum of T when D is of finite rank. As a second application, some consequences concerning the ergodic behavior of the operator T are derivedThe first author has been partly supported by Project PID2019-105979GB-I00, DGI-FEDER, of the MCYTS, Project E26-17R, D.G. Aragón, and Project for Young Researchers, Fundación Ibercaja and Universidad de Zaragoza, Spain. The second author has been partially supported by La Caixa-Severo Ochoa grant (ICMAT Severo Ochoa project SEV-2011-0087, Spanish Ministry of Economic Affairs and Digital Transformation (MINECO)). Both second and third authors acknowledge partial support by Spanish Ministry of Science, Innovation and Universities (grant no. PGC2018-099124-B-I00) and the ICMAT Severo Ochoa project SEV-2015-0554 of the Spanish Ministry of Economy and Competitiveness of Spain and the European Regional Development Fund, through the “Severo Ochoa Programme for Centres of Excellence in R&D”. Both second and third authors also acknowledge fi nancial support from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001

Trace inequalities for positive operators via recent refinements and reverses of Young's inequality

In this paper we obtain some trace inequalities for positive operators via recent refinements and reverses of Young’s inequality due to Kittaneh-Manasrah, Liao-Wu-Zhao, Zuo-Shi-Fujii, Tominaga and Furuichi

An estimate for the numerical radius of the Hilbert space operators and a numerical radius inequality

We provide a number of sharp inequalities involving the usual operator norms of Hilbert space operators and powers of the numerical radii. Based on the traditional convexity inequalities for nonnegative real numbers and some generalize earlier numerical radius inequalities, operator. Precisely, we prove that if \A_i,\B_i,\X_i\in\bh ($i=1,2,\cdots,n$), $m\in\N$, $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ and $\phi$ and $\psi$ are non-negative functions on $[0,\infty)$ which are continuous such that $\phi(t)\psi(t)=t$ for all $t \in [0,\infty)$, then \begin{equation*} w^{2r}\bra{\sum_{i=1}^{n}\X_i\A_i^m\B_i}\leq \frac{n^{2r-1}}{m}\sum_{j=1}^{m}\norm{\sum_{i=1}^{n}\frac{1}{p}S_{i,j}^{pr}+\frac{1}{q}T_{i,j}^{qr}}-r_0\inf_{\norm{x}=1}\rho(\xi), \end{equation*} where $r_0=\min\{\frac{1}{p},\frac{1}{q}\}$, S_{i,j}=\X_i\phi^2\bra{\abs{\A_i^{j*}}}\X_i^*, T_{i,j}=\bra{\A_i^{m-j}\B_i}^*\psi^2\bra{\abs{\A_i^j}}\A_i^{m-j}\B_i and \rho(x)=\frac{n^{2r-1}}{m}\sum_{j=1}^{m}\sum_{i=1}^{n}\bra{\seq{S_{i,j}^r\xi,\xi}^{\frac{p}{2}}-\seq{T_{i,j}^r\xi,\xi}^{\frac{q}{2}}}^2.Comment: No comment

Impossibility of dimension reduction in the nuclear norm

Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear norm $\|T\|_{\mathsf{S}_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values of $T$. We prove that for arbitrarily large $n\in \mathbb{N}$ there exists a subset $\mathcal{C}\subseteq \mathsf{S}_1$ with $|\mathcal{C}|=n$ that cannot be embedded with bi-Lipschitz distortion $O(1)$ into any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. $\mathcal{C}$ is not even a $O(1)$-Lipschitz quotient of any subset of any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$. Thus, $\mathsf{S}_1$ does not admit a dimension reduction result \'a la Johnson and Lindenstrauss (1984), which complements the work of Harrow, Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with $\mathsf{S}_1$ replaced by the Banach space $\ell_1$ of absolutely summable sequences via the work of Brinkman and Charikar (2003). In fact, the above set $\mathcal{C}$ can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in $\mathsf{S}_1$. The challenge is to demonstrate that $\mathcal{C}$ cannot be faithfully realized in an arbitrary low-dimensional subspace of $\mathsf{S}_1$, while Brinkman and Charikar obtained such an assertion only for subspaces of $\mathsf{S}_1$ that consist of diagonal operators (i.e., subspaces of $\ell_1$). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace $X$ of $\mathsf{S}_1$ is at most a universal constant multiple of $\sqrt{\log \mathrm{dim}(X)}$

The transmission problem on a three-dimensional wedge

We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge

K-spectral sets, operator tuples and related function theory

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 28-04-2017La investigación en la que se basa esta tesis ha sido financiada por los proyectos MTM2011-28149-C02-1 y MTM2015-66157-C2-1-P del Ministerio de Economía y Competitividad, cuyo investigador principal es José Luis Torrea