Inequalities for the Hadamard Weighted Geometric Mean of Positive Kernel Operators on Banach Function Spaces

Abstract

Let K1, . . . , Kn be positive kernel operators on a Banach function space. We prove that the Hadamard weighted geometric mean of K1, . . . , Kn, the operator K, satisfies the following inequalitieswhere · and r(·) denote the operator norm and the spectral radius, respectively.In the case of completely atomic measure space we show some additional results. In particular, we prove an infinite-dimensional extension of the known characterization of those functions f : IR n + → IR+ satisfying r(f (A1, . . . , An)) ≤ f (r(A1), . . . , r(An)) for all non-negative matrices A1, . . ., An of the same order.

Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces

“…As proved in [5] and [15], the inequalities in Theorem 2.1 and Corollary 2.2 can be extended to positive kernel operators on Banach function spaces provided…”

“…[7], [8], [5], [15], [6], [17], [16], [4]). It will also be one of the main tools in the current paper.…”

“…Since our first theorem in the next section gives an inequality for these general spaces, we shortly recall some basic definitions and results from [5] and [15].…”

“…Schep extended inequalities (1.1) and (1.2) to nonnegative matrices that define bounded operators on sequence spaces (in particular on l p spaces, 1 ≤ p < ∞). In the proofs certain results on the Hadamard product from [5] were used. It was claimed in [17 It turned out that ρ(AB • BA) and ρ(AB • AB) may in fact be different and that (1.5) is false in general.…”

“…In the second section we introduce some definitions and facts and recall some results from [5] and [15], which we will need in our proofs. In the third section we extend and/or refine several inequalities from [13], [16], [4], [5] and [15] (including the inequalities (1.3) and (1.8)) to non-negative matrices that define bounded operators on sequence spaces. More precisely, in Theorem 3.1 we prove a version of inequality (1.3), which is valid for arbitrary positive kernel operators on Banach function spaces.…”