Isometric composition operators on the analytic Besov spaces

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“…Here, we rely on a proof by a special flavor that is due to Allen et.al. in [1], so we omitted the details. One of the tactics is used for using a fact that adjoint of a surjective isometry is also an isometry with the fact that the point estimates are really bounded linear functions.…”

“…It goes without saying that W ψ,φ f is a generalization of C φ f = f • φ and multiplication operator M ψ f = ψ • f . The demeanor of these operators is extensively studied on numerous spaces of holomorphic functions (see for example [1,3,4,6,14,15,22] and others).…”

“…al. in [1], they showed that the isometries among C φ operators on the Besov spaces B p for 1 ≤ p < ∞ are induced by rotations. Shabazz and Tjani extended this work to all Besov type spaces B p,α for p > 1, α > −1 see [22].…”

Composition Operators on Some Banach Spaces of Harmonic Mappings

“…In this section, we wish to characterize the isometries on the harmonic spaces B α H , A − α H , Z H , and B p H . For most of the corresponding analytic counterparts, namely, B α for α ≠ 1, A − α , Z, and B p for p > 1 and p ≠ 2, the only composition operators that are isometries are those induced by rotations [20][21][22][23] (see also [24]). Since an isometry on a harmonic space X H that extends an analytic space X is also an isometry on X, Proof.…”

Isometries on Some General Family Function Spaces Among Composition Operators

“…It goes without saying that W ψ,φ f is a generalization of C φ f = f • φ and multiplication operator M ψ f = ψ • f . The demeanor of these operators is extensively studied on numerous spaces of holomorphic functions (see for example [1,3,4,6,14,15,22] and others).…”