162 research outputs found

    An operator extension of the parallelogram law and related norm inequalities

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    We establish a general operator parallelogram law concerning a characterization of inner product spaces, get an operator extension of Bohr's inequality and present several norm inequalities. More precisely, let A{\mathfrak A} be a Cβˆ—C^*-algebra, TT be a locally compact Hausdorff space equipped with a Radon measure ΞΌ\mu and let (At)t∈T(A_t)_{t\in T} be a continuous field of operators in A{\mathfrak A} such that the function t↦Att \mapsto A_t is norm continuous on TT and the function t↦βˆ₯Atβˆ₯t \mapsto \|A_t\| is integrable. If Ξ±:TΓ—Tβ†’C\alpha: T \times T \to \mathbb{C} is a measurable function such that Ξ±(t,s)Λ‰Ξ±(s,t)=1\bar{\alpha(t,s)}\alpha(s,t)=1 for all t,s∈Tt, s \in T, then we show that \begin{align*} \int_T\int_T&\left|\alpha(t,s) A_t-\alpha(s,t) A_s\right|^2d\mu(t)d\mu(s)+\int_T\int_T\left|\alpha(t,s) B_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) \nonumber &= 2\int_T\int_T\left|\alpha(t,s) A_t-\alpha(s,t) B_s\right|^2d\mu(t)d\mu(s) - 2\left|\int_T(A_t-B_t)d\mu(t)\right|^2\,. \end{align*}Comment: 9 pages; To appear in Math. Inequal. Appl. (MIA

    Hyers-Ulam-Rassias Stability of Generalized Derivations

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    The generalized Hyers--Ulam--Rassias stability of generalized derivations on unital Banach algebras into Banach bimodules is established.Comment: 9 pages, minor changes, to appear in Internat. J. Math. Math. Sc

    Matrix Hermite-Hadamard type inequalities

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    We present several matrix and operator inequalities of Hermite-Hadamard type. We first establish a majorization version for monotone convex functions on matrices. We then utilize the Mond-Pecaric method to get an operator version for convex functions. We also present some applications. Finally we obtain an Hermite-Hadamard inequality for operator convex functions, positive linear maps and operators acting on Hilbert spaces.Comment: 13 pages, revised versio

    On (Co)Homology of Triangular Banach Algebras

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    Suppose that A and B are unital Banach algebras with units 1_A and 1_B, respectively, M is a unital Banach A-B-bimodule, T=Tri(A,M,B) is the triangular Banach algebra, X is a unital T-bimodule, X_{AA}=1_AX1_A, X_{BB}=1_BX1_B, X_{AB}=1_AX1_B and X_{BA}=1_BX1_A. Applying two nice long exact sequences related to A, B, T, X, X_{AA}, X_{BB}, X_{AB} and X_{BA} we establish some results on (co)homology of triangular Banach algebras.Comment: 10 page
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