Isometries of some \(F\)-algebras of holomorphic functions
For \(p>1\) let \(N^p\) be the algebra of all holomorphic functions on the open unit disk \(\mathbb{D}\) satisfying \(\sup_{0<r <1}\int_T(\log (1+| f(r\zeta) |))^p d\sigma (\zeta) <\infty\). Here \(T\) is the unit circle and \(\sigma\) is the normalized Lebesgue measure on \(T\). Endowed with the metric \[ d(f,t)= \Biggl( \int_T \Bigl( \log\biggl( 1+\bigl| f^*(\zeta)-g^*(\zeta) \bigr| \biggr) \Bigr)^p d\sigma (\zeta) \Biggr)^{{1\over p}}, \] where \(f^*\) is the boundary function of \(f\), \(N^p\) becomes an \(F\)-algebra (see \textit{M. Stoll} [Ann. Pol. Math. 35, 139-158 (1977; Zbl 0377.30036)]). Note that \(N^p\) is a subset of the Smirnov class \(N_*\), that is the set of all holomorphic functions in \(\mathbb{D}\) for which the family \(\{\log (1+| f(r\zeta) |)\), \(0\leq r<1\}\) is uniformly integrable on \(T\). The authors show that a linear map \(A\) on \(N^p\) is an isometry of \(N^p\) onto \(N^p\) if and only if there exist inner functions \(\Psi\) and \(\Phi\), where \(\Phi^*\) is a measure preserving map on \(T\), such that \((Af)(z)= \Psi(z) f(\Phi (z))\), \(z\in \mathbb{D}\), \(f\in A\). Note that \(\Phi^*\) is measure preserving if and only if \(\Phi(0)=0\). Thus the isometries on \(N^p\) have the same structure as those on \(N_*\) (see \textit{K. Stephenson} [Indiana Univ. Math. J. 26, 307-324 (1977; Zbl 0353.30024)]). The proof, similar to that of K. Stephenson's, uses the fact, that the restriction to \(H^p\) of an isometry in \(N^p\) actually is an isometry on \(H^p\). Recall that the isometries on \(H^p\), \(p>0\), \(p\neq 2\) have been characterized by \textit{W. Rudin} [Indiana Univ. Math. J. 25, 215-228 (1976; Zbl 0326.46011)].
scientific article; zbMATH DE number 1229808