Inequalities on the spectral radius, operator norm and numerical radius of the Hadamard weighted geometric mean of positive kernel operators. Issue 8 (3rd August 2019)

• Record Type: Journal Article
• Authors: Peperko, Aljoša
• Is Part Of: Linear & multilinear algebra. Volume 67:Issue 8(2019)
• Page Start: 1637
• Page End: 1652
• Publication Date: 2019-08-03
• Subjects: Hadamard-Schur weighted geometric mean -- spectral radius -- positive kernel operators -- Banach function spaces -- nonnegative matrices; Primary: 47B65 -- Secondary: 47A10 -- 47B34 -- 15A60; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1465885
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: Recently, several authors have proved inequalities on the spectral radius ρ, operator norm ‖ · ‖ and numerical radius of Hadamard products and ordinary products of nonnegative matrices that define operators on sequence spaces, or of the Hadamard geometric mean and ordinary products of positive kernel operators on Banach function spaces. In the present article we generalize and refine several of these results. In particular, we show that for a Hadamard geometric mean A 1 2 ∘ B 1 2 of positive kernel operators A and B on a Banach function space L, we have ρ A ( 1 2 ) ∘ B ( 1 2 ) ≤ ρ 1 2 ( A B ) ( 1 2 ) ∘ ( B A ) ( 1 2 ) ≤ ρ 1 2 ( A B ) . In the special case L = L 2 ( X, μ ) we also prove that ‖ A ( 1 2 ) ∘ B ( 1 2 ) ‖ ≤ ρ 1 2 ( A ∗ B ) ( 1 2 ) ∘ ( B ∗ A ) ( 1 2 ) ≤ ρ 1 2 ( A ∗ B ) .