On characterizations of isometries on function spaces (Q943411)

\nopagenumbers Let \(X\) be a complex Banach space. An isometric isomorphism on \(X\) is an onto linear operator \(T:X \rightarrow X\) such that \(| Tx| =| x| \) for all \(x \in X\). In the present article, the isometric isomorphisms of special spaces of holomorphic functions over the unit ball \(B_n\) in \(\mathbb C^n\) are characterized. The first is the little Bloch space \[ {\mathcal B}^0(B_n) := \{ f :B_n \longrightarrow \mathbb C: Rf(z)(1-| z| ^2) \rightarrow 0,\text{ as } z \to \partial B_n\} \] equipped with the Bloch norm \[ \| f\| _{{\mathcal B}^0(B_n)}:= \sup \{ | Rf(z)| (1-| z| ^2):z \in B_n\}. \] Here, \(Rf(z):= \sum_{j=1}^n z_j \frac{\partial f}{\partial z_j}(z) \). Theorem: An operator \(T: {\mathcal B}^0(B_n) \rightarrow {\mathcal B}^0(B_n)\) is an isometric isomorphism if and only if there exists a unitary matrix \(U\) and a number \(\theta \in [0, 2\pi)\), such that \[ Tf(z) = e^{i\theta} f(Uz) ,\qquad \forall f \in {\mathcal B}^0(B_n),\quad z \in B_n. \] For a bounded domain \(D\), let \(P:L^2(D) \rightarrow A^2(D)\) denote the Bergman projector. If \({\mathcal B}(D)\) denotes the Bloch space over \(D\), then \(P(L^\infty(D))={\mathcal B}(D)\). Now, let for \(f \in {\mathcal B}(D)\): \[ \| f\| _*:= \inf\{ \| g\| _{L^\infty(D)}:P(g)=f\}. \] Then the space \({\mathcal B}^0(B_n)\), equipped with the norm \(\| \cdot \| _*\), becomes a Banach space that is denoted by \({\mathcal B}^{\infty, -}(B_n)\). The isometric isomorphisms of \({\mathcal B}^{\infty, -}(B_n)\) are also characterized. Likewise, the isometric isomorphisms of the space VMOA of vanishing mean oscillation over \(B_n\) and the Besov spaces are completely determined.