Inequalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators. Issue 11 (2nd November 2019)

• Record Type: Journal Article
• Authors: Peperko, Aljoša
• Is Part Of: Linear & multilinear algebra. Volume 67:Issue 11(2019)
• Page Start: 2159
• Page End: 2172
• Publication Date: 2019-11-02
• Subjects: Hadamard–Schur geometric mean -- Hadamard–Schur product -- joint and generalized spectral radius -- essential spectral radius -- measure of noncompactness -- positive kernel operators -- non-negative matrices -- bounded sets of operators; 15A42 -- 15A60 -- 47B65 -- 47B34 -- 47A10 -- 15B48; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1484423
• Languages: English
• ISSNs: 0308-1087

Abstract

ABSTRACT: Let Ψ and Σ be bounded sets of positive kernel operators on a Banach function space L . We prove several refinements of the known inequalities ρ Ψ ( 1 / 2 ) ∘ Σ ( 1 / 2 ) ≤ ρ ( ΨΣ ) 1 / 2 and ρ ˆ Ψ ( 1 / 2 ) ∘ Σ 1 / 2 ≤ ρ ˆ ( ΨΣ ) 1 / 2 for the generalized spectral radius ρ and the joint spectral radius ρ ˆ, where Ψ ( 1 / 2 ) ∘ Σ ( 1 / 2 ) denotes the Hadamard (Schur) geometric mean of the sets Ψ and Σ. Furthermore, we prove that analogous inequalities hold also for the generalized essential spectral radius and the joint essential spectral radius in the case when L and its Banach dual L ∗ have order continuous norms.