Bounds on the generalized and the joint spectral radius of Hadamard products of bounded sets of positive operators on sequence spaces

Abstract

Let Ψ 1 ,. .. Ψ m be bounded sets of positive kernel operators on a Banach function space L. We prove that for the generalized spectral radius ρ and the joint spectral radiusˆρradiusˆ radiusˆρ the inequalities ρ Ψ (1 m) 1 • · · · • Ψ (1 m) m ≤ ρ(Ψ 1 Ψ 2 · · · Ψ m) 1 m , ˆ ρ Ψ (1 m) 1 • · · · • Ψ (1 m) m ≤ ˆ ρ(Ψ 1 Ψ 2 · · · Ψ m) 1 m hold, where Ψ (1 m) 1 • · · · • Ψ (1 m) m denotes the Hadamard (Schur) geometric mean of the sets Ψ 1 ,. .. , Ψ m .

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

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

“…the proof of [16,Theorem 3.7]). Moreover, an easy application of Corollary 2.2 gives the following infinite-dimensional generalization of (3.8) and its analogues for the operator norm and the numerical radius.…”

“…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. This typing error was corrected in [18] and [16]. Moreover, it was proved in [16] that for non-negative matrices that define bounded operators on sequence spaces the inequalities and (1.3) hold.…”

“…This typing error was corrected in [18] and [16]. Moreover, it was proved in [16] that for non-negative matrices that define bounded operators on sequence spaces the inequalities and (1.3) hold.…”

“…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.…”