If $A,B \in \mathcal {B}(\mathcal {H})$ are normal contractions, then for every $X \in \mathcal {C}_{||| \cdot |||}(\mathcal {H})$ and $\alpha > 0$ holds $$\Big|\Big|\Big| (I - A^{*}A)^{\frac{\alpha}{2}} X(I - B^{*}B)^{\frac{\alpha}{2}} \Big|\Big|\Big| \leqslant \Big|\Big|\Big| \sum_{n=0}^\infty (-1)^{n} \binom{\alpha}{n}A^{n} X B^{n} \Big|\Big|\Big|,$$ which generalizes a result of D.R. Jocić [Proc. Amer. Math. Soc. 126 (1998), no. 9, 2705-2713] for $\alpha$ not being an integer. Similar inequalities in the Schatten $p$-norms, for non-normal $A,B$ and in the $Q$-norms if one of $A$ or $B$ is normal, are also given.

H. Dye proved that the discrete unitary group in a factor determines the algebraic type of the factor. Afterwards, for a large class of simple unital $C^{*}$-algebras, Al-Rawashdeh, Booth and Giordano proved that the algebras are $*$-isomorphic if and only if their unitary groups are isomomorphic as abstract groups. In this paper, we give a counter example in the non-simple case. Indeed, we give two $C^{*}$-algebras with isomorphic unitary groups but the algebras themselves are not $*$-isomorphic.

In this paper, we introduce a new algorithm for finding a common fixed point of finite family of continuous pseudocontractive mappings which is a unique solution of somevariational inequality problem and whose image under some bounded linear operator isa common solution of some system of equilibrium problems in a real Hilbert space. Ourresult generalize and improve some well-known results.

We refine the Hölder-McCarthy inequality. The point is the convexity of the function induced by Hölder-McCarthy inequality. Also we discuss the equivalent between refined Hölder-McCarthy inequality and refined Young inequality with type of Kittaneh and Manasrah.

This paper studies the generalized approximate set-valued equilibrium problems and furnishes some new existence results. The existence results for solutions are derived by using the celebrated KKM theorem and some concepts associated with the semi-continuity of the set-valued mappings such as outer-semicontinuity, inner-semicontinuity, upper-semicontinuity and so forth. The results achieved in this paper generalize and improve the works of many authors in references.

In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs. The construction is based on a specific tessellation on the considered graph, it allows us to express the Markov property for the local structure of the graph. Our main result is the existence and uniqueness of quantum Markov field.

Tsallis relative operator entropy was firstly formulated by Fujii and Kamei as an operator version of Uhlmann's relative entropy. Afterwards, Yanagi, Kuriyama and Furuichi reformulated Tsallis relative operator entropy as an operator version of Tsallis relative entropy. In this paper, we define Tsallis relative operator entropy with negative parameters of (non-invertible) positive operators on a Hilbert space and show some properties.