In this paper, we give the Alzer inequality for Hilbert space operators as follows:Let A, B be two selfadjoint operators on a Hilbert space H such that 02 are arithmetic and geometric means of A, B, respectively, where 0 < λ < 1. We show that if A and B are commuting, thenwhere A ′ := I − A, B ′ := I − B and 0 < λ ≤ 1 2 . Also, we state an open problem for an extension of Alzer inequality.
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 A = (A1, . . . , An) and B = (B1, . . . , Bn) 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 setting
Abstract. The theorems of Bernstein-Doetsch, and Ostrowski, concerning the continuity of midconvex functions are extended to open subsets of locally compact and root-approximable topological groups.
Abstract. Let A, B and C be adjointable operators on a Hilbert C * -module 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 + Y B = 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 * + BY B * = C. Finally, we deduce that there exist nonzero operators X, Y ≥ 0 and Z such that AXA * + BY B * = CZ, when A, B and C are given subject to some conditions.
Introduction and PreliminariesRecently several operator equations have been extended from matrices to infinite dimensional spaces, i.e., Hilbert spaces and Hilbert C * -modules; see [11] and references therein. Recall that the notion of Hilbert C * -module is a natural generalization of that of Hilbert space arising by replacing the field of scalars C by a C * -algebra. Generalized inverses are useful tools for investigation of solutions of operator equations in the setting of Hilbert C * -modules but these inverses need the strong condition of closedness of ranges of considered operators. Fang et al. [6,7] have studied the solvability of operator equations without the closedness condition on ranges of operators by employing a generalization of a known theorem of Douglas [4, Theorem 1] in the framework of Hilbert C * -modules. In their results, concentration is based on the idea of using more general (orthogonal) projections instead of projections such as AA † . They investigated the equations AX = B Inspired by Fang et al., we investigate the solution of equations AX + Y B = C and AXA * +BY B * = C without the condition of closedness of ranges. This paper is organized as follows. First, we recall some basic information about Hilbert C * -modules. In Section 2, we present an example that shows that the conditions CC * ≤ λAA * for some λ > 0 and R(C) ⊆ R(A) are not equivalent in the setting 2010 Mathematics Subject Classification. 15A24, 46L08, 47A05, 47A62.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.