On continuous composition operators

• Autorzy: Wilhelmina Smajdor
• Języki publikacji: EN

Abstrakty

EN

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 𝓐(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.

Kategorie tematyczne

• 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytski{\u\i}, Uryson, etc.)
• 47B38: Operators on function spaces (general)
• 39B52: Equations for functions with more general domains and/or ranges

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2010
• Tom: 98
• Numer: 3
• Strony: 273-282

Daty

wydano

2010

Twórcy

autor

Wilhelmina Smajdor

• Technical University of Technology, Kaszubska 23, PL-44-100 Gliwice, Poland

Identyfikatory

DOI

10.4064/ap98-3-6