Submultiplicative functions and operator inequalities

Hermann König, and Vitali Milman. "Submultiplicative functions and operator inequalities." Studia Mathematica 223.3 (2014): 217-231. <http://eudml.org/doc/285878>.

@article{HermannKönig2014,
abstract = {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 = ⎧ $(H∘f/H)f^\{\prime p\}$, f’ ≥ 0, ⎨ ⎩ $-A(H∘f/H)|f^\{\prime \}|^\{p\}$, 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.},
author = {Hermann König, Vitali Milman},
journal = {Studia Mathematica},
keywords = {submultiplicative functions; chain rule; operator inequalities},
language = {eng},
number = {3},
pages = {217-231},
title = {Submultiplicative functions and operator inequalities},
url = {http://eudml.org/doc/285878},
volume = {223},
year = {2014},
}

TY - JOUR
AU - Hermann König
AU - Vitali Milman
TI - Submultiplicative functions and operator inequalities
JO - Studia Mathematica
PY - 2014
VL - 223
IS - 3
SP - 217
EP - 231
AB - 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 = ⎧ $(H∘f/H)f^{\prime p}$, f’ ≥ 0, ⎨ ⎩ $-A(H∘f/H)|f^{\prime }|^{p}$, 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.
LA - eng
KW - submultiplicative functions; chain rule; operator inequalities
UR - http://eudml.org/doc/285878
ER -