On continuous composition operators

Wilhelmina Smajdor. "On continuous composition operators." Annales Polonici Mathematici 98.3 (2010): 273-282. <http://eudml.org/doc/281100>.

@article{WilhelminaSmajdor2010,
abstract = {Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that $M_\{ψ\}:= sup\{||[r,s,t;ψ]||: r < s < t, r,s,t ∈ I\} < ∞$, where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove that if N is continuous and $M_\{ψ\} ≤ M$ for some constant M > 0, where ψ(t) = N(t,φ(t)), then h(t,y) = a(y) + b(t), t ∈ I, y ∈ Y, for some continuous linear a: Y → Z and b ∈ Lip₂(I,Z). Lipschitzian and Hölder composition operators are also investigated.},
author = {Wilhelmina Smajdor},
journal = {Annales Polonici Mathematici},
keywords = {divided differences; functions with bounded divided differences; functions with Lipschitzian derivatives; Lipschitz and Hölder conditions},
language = {eng},
number = {3},
pages = {273-282},
title = {On continuous composition operators},
url = {http://eudml.org/doc/281100},
volume = {98},
year = {2010},
}

TY - JOUR
AU - Wilhelmina Smajdor
TI - On continuous composition operators
JO - Annales Polonici Mathematici
PY - 2010
VL - 98
IS - 3
SP - 273
EP - 282
AB - Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that $M_{ψ}:= sup{||[r,s,t;ψ]||: r < s < t, r,s,t ∈ I} < ∞$, where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove that if N is continuous and $M_{ψ} ≤ M$ for some constant M > 0, where ψ(t) = N(t,φ(t)), then h(t,y) = a(y) + b(t), t ∈ I, y ∈ Y, for some continuous linear a: Y → Z and b ∈ Lip₂(I,Z). Lipschitzian and Hölder composition operators are also investigated.
LA - eng
KW - divided differences; functions with bounded divided differences; functions with Lipschitzian derivatives; Lipschitz and Hölder conditions
UR - http://eudml.org/doc/281100
ER -