A new approach to γ-bounded representations

2024/02/16 by Christian Le Merdy, Merdy, Christian Le
Engineering · Computer Science · Mathematics · #Fault Detection and Control Systems #Rough Sets and Fuzzy Logic #Mathematical Analysis and Transform Methods

paper · pdf · doi:10.48550/arxiv.2402.10736

Abstract

Let X be a Banach space, let (Ω,μ) be a σ-finite measure space and let A,B\colonΩ→ B(X) be strongly measurable γ-bounded functions. We show that for all x∈ X and all x^*∈ X^*, there exist a Hilbert space K and two measurable functions a₁∈ L^∞(Ω;K) and a₂∈ L^∞(Ω;K) such that ⟨ B(t)A(s)x,x^*⟩ = (a₂(t) \vert a₁(s))_K for a.e. (s,t) in Ω², with \Vert a₁\Vert_∞ \Vert a₂\Vert_∞≤ γ(A)γ(B)\Vert x\vert\vert x^*\Vert. This factorization property allows us to improve or simplify some results concerning γ-bounded representations of groups or semigroups.

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