An operator extension of the parallelogram law and related norm inequalities

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 ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a Radon measure $\mu$ and let $(A_t)_{t\in T}$ be a continuous field of operators in ${\mathfrak A}$ such that the function $t \mapsto A_t$ is norm continuous on $T$ and the function $t \mapsto \|A_t\|$ is integrable. If $\alpha: T \times T \to \mathbb{C}$ is a measurable function such that $\bar{\alpha(t,s)}\alpha(s,t)=1$ for all $t, s \in T$, 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

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

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

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