arXiv subjects
Publications and source records attributed to F. Mirzapour.
In this paper, we will derive the real roots of certain sets of matrices with real entries. We will also demonstrate that real orthogonal matrices can have real root or be involutory. Eventually, we will represent idempotent matrices in a block form.
In this paper, we state some characterizations of $h$-convex function is defined on a convex set in a linear space. By doing so, we extend the Jensen-Mercer inequality for $h$-convex function. We will also define $h$-convex function for operators on a Hilbert space and present the operator version of the Jensen-Mercer inequality. Lastly, we propound the complementary inequality of Jensen's inequality for $h$-convex functions.
Let $A,B$ and $C$ be adjointable operators on a Hilbert $C^*$-module $\mathscr{E}$. Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert $C^*$-modules, we present the general solution of the equation $AX+YB=C$ when the ranges of $A,B$ and $C$ are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert $C^*$-modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for $AXA^*+BYB^*=C$. Finally, we deduce that there exist nonzero operators $X, Y\geq 0$ and $Z$ such that $AXA^*+BYB^*=CZ$, when $A, B$ and $C$ are given subject to some conditions.
We establish some operator versions of Bellman's inequality. In particular, we prove that if $Φ: \mathbb{B}(\mathscr{H}) \to \mathbb{B}(\mathscr{K})$ is a unital positive linear map, $A,B \in \mathbb{B}(\mathscr{H})$ are contractions, $p>1$ and $0 \leq λ\leq 1$, then {eqnarray*} \big(Φ(I_\mathscr{H}-A\nabla_λB)\big)^{1/p}\geΦ\big((I_\mathscr{H}-A)^{1/p}\nabla_λ(I_\mathscr{H}-B)^{1/p}\big). {eqnarray*}
In this paper we investigate a notion of relative operator entropy, which develops the theory started by J.I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341--348]. For two finite sequences $\mathbf{A}=(A_1,...,A_n)$ and $\mathbf{B}=(B_1,...,B_n)$ of positive operators acting on a Hilbert space, a real number $q$ and an operator monotone function $f$ we extend the concept of entropy by $$ S_q^f(\mathbf{A}|\mathbf{B}):=\sum_{j=1}^nA_j^{1/2}(A_j^{-1/2}B_jA_j^{-1/2})^qf(A_j^{-1/2}B_jA_j^{-1/2})A_j^{1/2}\,, $$ and then give upper and lower bounds for $S_q^f(\mathbf{A}|\mathbf{B})$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219--235] under certain conditions. Afterwards, some inequalities concerning the classical Shannon entropy are drawn from it.