Isometries among composition operators on Besov type spaces

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.…”

Closed range properties of Li–Stević integral-type operators between Bloch-typespaces and their essential norms

“…See [24] for the above mentioned definitions and note that Bloch-type spaces B v are examples of Banach spaces X with norm satisfying conditions (3.1) and (3.2). At the end of this section we will give more examples of such Banach spaces X .…”

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“…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).…”