Around Operator Monotone Functions

Abstract

Abstract. We show that the symmetrized product AB + BA of two positive operators A and B is positive if and only if f (A + B) ≤ f (A) + f (B) for all non-negative operator monotone functions f on [0, ∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition f • g of an operator convex function f on [0, ∞) and a non-negative operator monotone function g on an interval (a, b) is operator monotone and give some applications.

On the binary relation⩽uon self-adjoint Hilbert space operators

“…The following result is a variant of Theorem 2.6 of [10]. If f is an increasing operator convex function, then f ′ + (0) ≥ 0.…”

“…The antisymmetric law states that Hilbert spaces. Utilizing a result of[10] we show that A ≤ u B if and only if f (g(A) r ) ≤ u f (g(B) r ) for any increasing operator convex function f , any operator monotone function g and any positive number r. Recall that a real function f defined on an interval J is said to be operator convex if f (λA + (1 − λ)B) ≤ λf (A) + (1 − λ)f (B) for any A, B ∈ B h (H ) with spectra in J and λ ∈ [0, 1] and is called operator monotone if f (A) ≤ f (B) whenever A ≤ B for any A, B ∈ B h (H ) with spectra in J, see…”

The Hermite-Hadamard Type Inequalities for Operator s-Convex Functions

“…In [10], Moslehian and Najafi proved the following theorem for positive operators as follows: As an example of operator s-convex function, we give the following example.…”

Approximation by (p, q)-Baskakov–Beta operators

“…Approximation theory has been an engaging field of research with abstract approximation to the core (cf. [15]). Varied operators with their approximation properties, mainly the quantitative one, have been discussed and studied by many researchers.…”