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.
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 .
have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite non-negative matrices that define operators on sequence spaces. In the current paper we extend and refine several of these results and also prove some analogues for the numerical radius. Some inequalities seem to be new even in the case of n × n non-negative matrices.
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