Reflexivity on weighted Hardy spaces.(English)Zbl 1081.47039

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 D. A. Herrero and A. Lambert [Trans. Am. Math. Soc. 185, 229–235 (1974; Zbl 0253.46127), Corollary 5].

MSC:

 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47L10 Algebras of operators on Banach spaces and other topological linear spaces

Zbl 0253.46127