Abstract. A Hermite-Hadamard-Mercer type inequality is presented and then generalized to Hilbert space operators. It is shown that. . , n, and A i are positive operators acting on a finite dimensional Hilbert space whose sum is equal to the identity operator. A Jensen-Mercer operator type inequality for separately operator convex functions is also presented.
We review some significant generalizations and applications of the celebrated Douglas theorem on equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of 2 × 2 block matrices of operators on Hilbert spaces and finally describe the nature of such block matrices and provide several ways for showing their positivity.
There exist two major subclasses in the class of superquadratic functions, one contains concave and decreasing functions and the other, contains convex and monotone increasing functions. We use this fact and present eigenvalue inequalities in each case. The nature of these functions enables us to apply our results in two directions. First improving some known results concerning eigenvalues for convex functions and second, obtaining some complimentary inequalities for some other functions. We will give some examples to support our results as well. As applications, some subadditivity inequalities for matrix power functions have been presented. In particular, if X and Y are positive matrices, thenfor some unitaries U, V ∈ M n .
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