On continuous composition operators

Wilhelmina Smajdor

Annales Polonici Mathematici, Tome 98 (2010), p. 273-282 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2010-01-01

Zbl 1203.47032

EUDML-ID : urn:eudml:doc:281100