Reflexivity on weighted Hardy spaces
For a sequence of positive integers \(\{\beta(n)\}^\infty_{n=-\infty}\) (resp., \(\{B(n)\}^\infty_{n=0})\) such that \(\beta(0) = 1\) and for \(1\leq p<\infty\), let \(L^p(\beta)\) (resp., \(H^p(\beta)\)) denote the Banach space of all formal power series \(f(z)=\sum^\infty_{n=-\infty}\widehat f(n)z^n\) (resp., \(f(z)=\sum^\infty_{n=0}\widehat f(n)z^n)\) such that the respective norm \(\| f\|_\beta\equiv(\sum_n|\widehat f(n)|^p\) \(\beta(n)^p\)) is finite. In this paper, the authors first prove in Corollary 2.2 that (i) the commutant of \(M_z\), the multiplication operator by the variable \(z\) on \(H^p(\beta)\), equals the weak closure of polynomials in \(M_z\), (ii) the same holds if \(M_z\) is not invertible on \(L^p(\beta)\), and (iii) the commutant of \(M_z\) equals the weak closure of rational functions, with poles off the spectrum of \(M_z\), in \(M_z\) if \(M_z\) is invertible on \(L^p(\beta)\). These are analogues of the known results for \(p = 2\). The paper then concludes with some results on the reflexivity of \(M_z\). One of them, Theorem 2.4, is known to be true under more general conditions when \(p = 2\), namely, every invertible \(M_z\) on \(L^2(\beta)\) is reflexive as was proved by \textit{D. A. Herrero} and \textit{A. Lambert} [Trans. Am. Math. Soc. 185, 229--235 (1974; Zbl 0253.46127), Corollary 5].
scientific article; zbMATH DE number 2144219