Related papers: Some extensions of the operator entropy type inequ…
Let $\alpha>0$ and $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu,\alpha}=(\mu_{n,k,\alpha})_{n,k\ge0}$ with entries…
Matrix concentration inequalities, intimately connected to the Non-Commutative Khintchine inequality, have been an important tool in both applied and pure mathematics. We study tensor versions of these inequalities, and establish…
We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known…
Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors establish an interpolation result that a sublinear operator which…
In this paper we improve Bernoulli comparison. The result works for independent Rademacher random variables $(\varepsilon_i)_{i\geq1}$ and states that we can compare $\mathbb{E}\sup_{t\in T}\sum_{i\geq1}\varphi_{i}(t)\varepsilon_i$ with…
A considerable amount of literature in the theory of inequality is devoted to the study of Jensen's and Young's inequality. This article presents a number of new inequalities involving the log-convex functions and the geometrically convex…
We prove an inequality for the entropy numbers in terms of nonlinear Kolmogorov's widths. This inequality is in a spirit of known inequalities of this type and it is adjusted to the form convenient in applications for $m$-term…
Matrix concentration inequalities provide a direct way to bound the typical spectral norm of a random matrix. The methods for establishing these results often parallel classical arguments, such as the Laplace transform method. This work…
The paper describes various approaches to the invertibility of Toeplitz plus Hankel operators in Hardy and $l^p$-spaces, integral and difference Wiener-Hopf plus Hankel operators and generalized Toeplitz plus Hankel operators. Special…
We propose an extension of the sandwiched R\'enyi relative $\alpha$-entropy to normal positive functionals on arbitrary von Neumann algebras, for the values $\alpha>1$. For this, we use Kosaki's definition of noncommutative $L_p$-spaces…
\begin{abstract} Let $P\pm$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider estimates of the expression $\|( |P_ + f | ^s + |P_- f |^s) ^{\frac{1}{s}}\|_{L^p (\mathbf{T})}$ in terms of Lebesgue $p$-norm of the…
A way to pose the entropic uncertainty principle for trace-preserving super-operators is presented. It is based on the notion of extremal unraveling of a super-operator. For given input state, different effects of each unraveling result in…
Tsallis relative operator entropy was defined as a parametric extension of relative operator entropy and the generalized Shannon inequalities were shown in the previous paper. After the review of some fundamental properties of Tsallis…
For a probability measure $\mu$ and for square integrable fields $(\mathscr{A}_t)$ and $(\mathscr{B}_t)$ ($t\in\Omega$) of commuting normal operators we prove Landau type inequality \llu\int_\Omega\mathscr{A}_tX\mathscr{B}_td\mu(t)-…
Tsallis relative operator entropy is defined and then its properties are given. Shannon inequality and its reverse one in Hilbert space operators derived by T.Furuta \cite{Fu:par} are extended in terms of the parameter of the Tsallis…
We consider the following trace function on n-tuples of positive operators: \Phi_p(A_1,A_2,...,A_n) = Trace (\sum_{j=1}^n A_j^p)^{1/p} and prove that it is jointly concave for 0<p\le 1 and convex for p=2. We then derive from this a…
The Adler-Konheim-McAndrew type definitions and the Bowen-Dinaburg-Hood type definitions of parametric topological entropy will be considered on orbits and coincidence orbits of nonautonomous multivalued maps in compact Hausdorff spaces.…
Let $ \tau : [0,1] \rightarrow [0,1] $ be a piecewise expanding map with full branches. Given $ \lambda : [0,1] \rightarrow (0,1) $ and $ g : [0,1] \rightarrow \mathbb{R} $ satisfying $ \tau ' \lambda > 1 $, we study the Weierstrass-type…
The paper explores three known methods, their variants and limitations, that can be used to obtain new entropy inequalities. The Copy Lemma was distilled from the original Zhang-Yeung construction which produced the first non-Shannon…
We study the following version of Hardy-type inequality on a domain $\Omega$ in a Riemannian manifold $(M,g)$: $$ \int{\Omega}|\nabla u|_g^p\rho^\alpha dV_g \geq \left(\frac{|p-1+\beta|}{p}\right)^p\int{\Omega}\frac{|u|^p|\nabla…