Some Jensen’s type inequalities for twice differentiable functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided.

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 [4] are extended in terms of the parameter of the Tsallis relative operator entropy. Moreover the generalized Tsallis relative operator entropy is introduced and then several operator inequalities are derived.

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 [5] are extended in terms of the parameter of the Tsallis relative operator entropy. Moreover the generalized Tsallis relative operator entropy is introduced and then several operator inequalities are derived.

The chaotic order A B among positive invertible operators A,B > 0 on a Hilbert space is introduced by logA≥ logB. Using Uchiyama’s method and Furuta’s Kantorovich-type inequality, we will point out that A B if and only if ‖BpA−p/2B−p/2‖Ap ≥ Bp holds for any 0 < p < p0, where p0 is any fixed positive number. On the other hand, for any fixed p0 > 0, we also show that there exist positive invertib...

Some norm inequalities for sequences of linear operators defined on Hilbert spaces that are related to the classical Schwarz inequality are given. Applications for vector inequalities are also provided.

The aim of this paper is to introduce the generalized viscosity implicit rules of one nonexpansive mapping in uniformly smooth Banach spaces. Strong convergence theorems of the rules are proved under certain assumptions imposed on the parameters. As applications, we use our main results to solve fixed point problems of strict pseudocontractions in Hilbert spaces and variational inequality probl...

In this paper, interpolating curve or surface with linear inequality constraints is considered as a general convex optimization problem in a Reproducing Kernel Hilbert Space. We propose a new approximation method based on a discretized optimization problem in a finite-dimensional Hilbert space under the same set of constraints. We prove that the approximate solution converges uniformly to the o...

We will show that for any two bounded linear operators X, Y on a Hilbert space H, if they satisfy the triangle equality |X + Y | = |X | + |Y |, there exists a partial isometry U on H such that X = U |X | and Y = U |Y |. This is a generalization of Thompson’s theorem to the matrix case proved by using a trace. 47A05, 47A10, 47A12 operator theory, Hilbert space, triangle inequality

where the constant factor [π/ sin(π/p)]p is also the best possible. Hardy-Hilbert inequalities are important in analysis and in their applications (see [7]). In recent years, many results (see [1, 3, 8–10]) have been obtained in the research of Hardy-Hilbert inequality. At present, because of the requirement of higher-dimensional harmonic analysis and higher-dimensional operator theory, multipl...

The main aim of the present paper is to establish various sharp upper bounds for the Euclidean operator radius of an n-tuple of bounded linear operators on a Hilbert space. The tools used are provided by several generalizations of Bessel inequality due to Boas-Bellman, Bombieri, and the author. Natural applications for the norm and the numerical radius of bounded linear operators on Hilbert spa...