Bessel- And Grüss-Type Inequalities in Inner Product Modules

Abstract

In this paper we give Bessel-and Grüss-type inequalities in an inner product module over a proper H * -algebra or a C * -algebra.

Landau and Grüss type inequalities for inner product type integral transformers in norm ideals

“…, associated to a (2-reconvexized) s.g function Φ (2) . Therefore the normability and the completeness of the space L 2 G (Ω, dµ, C Φ (H)), as stated in Theorem 2.1 in [13], follow immediately.…”

“…Take Φ to be a s.g. function that generates u.i. norm ||| · |||, Φ 1 = Φ, Φ 2 = Φ 3 = Φ (2) (2-reconvexization of Φ), α = 2θ and X = I, and then apply 3.1. (e) from [13].…”

“…It has been the subject of intensive investigation, in which conditions on functions are varied to obtain different estimates; see [8,20] and references therein. This inequality has been investigated, applied and generalized by many authors in different areas of mathematics, among others in inner product spaces [7], quadrature formulae [5,25], finite Fourier transforms [4], linear functionals [1,12], matrix traces [24], inner product modules over H * -algebras and C * -algebras [2,11],…”

Grüss type inequalities for positive linear maps on C∗-algebras

“…Renaud [17] showed that tr(T AB) − tr(T A)tr(T B) ≤ krs, (1.3) for 1 ≤ k ≤ 4, where A, B are square matrices, whose numerical ranges lie in the circular discs of radii r and s, respectively, and T is a positive matrix of trace one. The Grüss inequality was generalized in the setting of inner product spaces by Dragomir [7], and in the framework of inner product modules over H * -algebras and C * -algebras by Ilišević et al [1,9].…”

“…The Grüss inequality was generalized in the setting of inner product spaces by Dragomir [7], and in the framework of inner product modules over H * -algebras and C * -algebras by Ilišević et al [1,9].…”