Submultiplicative functions and operator inequalities

• Autorzy: Hermann König , Vitali Milman
• Języki publikacji: EN

Abstrakty

EN

Let T: C¹(ℝ) → C(ℝ) be an operator satisfying the "chain rule inequality"
T(f∘g) ≤ (Tf)∘g⋅Tg, f,g ∈ C¹(ℝ).
Imposing a weak continuity and a non-degeneracy condition on T, we determine the form of all maps T satisfying this inequality together with T(-Id)(0) < 0. They have the form
Tf =
⎧ $(H∘f/H)f'^{p}$, f' ≥ 0,
⎨
⎩ $-A(H∘f/H)|f'|^{p}$, f' < 0,
with p > 0, H ∈ C(ℝ), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable functions K on ℝ which are continuous at 0 and 1 and satisfy K(-1) < 0 < K(1). Any such map K has the form
K(α) =
⎧ $α^{p}$, α ≥ 0,
⎨
⎩ $-A|α|^{p}$, α < 0,
with A ≥ 1 and p > 0. Corresponding statements hold in the supermultiplicative case with 0 < A ≤ 1.

Kategorie tematyczne

• 46E15: Banach spaces of continuous, differentiable or analytic functions
• 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems
• 39B42: Matrix and operator equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2014
• Tom: 223
• Numer: 3
• Strony: 217-231

Daty

wydano

2014

Twórcy

autor

Hermann König

• Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany

autor

Vitali Milman

• School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Identyfikatory

DOI

10.4064/sm223-3-3