On the eighteenth question of Allen Shields. (English) Zbl 1081.47038

The eighteenth question in A. L. Shields’ survey paper [Math. Surveys 13, 49–128 (1974; Zbl 0303.47021)] asks for a characterization of reflexive (unilateral or bilateral) weighted shifts on \(\ell^p\) \((1\leq p<\infty)\). Note that injective bilateral (resp., unilateral) weighted shifts are, under unitary equivalence for \(p = 2\), representable as the multiplication operator \(M_z\) by the variable \(z\) on the Banach space \(L^p(\beta)\) (resp., \(H^p(\beta)\)) associated with a sequence \(\{\beta(n)\}^\infty_{n=-\infty}\) (resp., \(\{\beta(n)\}^\infty_{n=0})\) of positive numbers satisfying \(\beta(0) = 1\), which consists of 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 \(\| f\|_\beta\equiv (\sum_n|\widehat f(n)|^p\beta(n)^p)^{1/p}\) is finite. In the present paper, several sufficient conditions are given in order that \(M_z\) be reflexive, meaning that the only operators which leave invariant all the invariant subspaces of \(M_z\) are those in the weakly closed unital algebra generated by \(M_z\).
However, all the results here (at least for the case \(p=2\)) follow from more general ones in the literature. For example, it was shown by D. A. Herrero and A. Lambert [Trans. Am. Math. Soc. 185 (1973), 229–235 (1974; Zbl 0253.46127), Corollary 5] that every invertible bilateral weighted shift on \(\ell\) is reflexive. This covers Theorems 2.1, 2.2 and 2.4. Also, if \(T\) is a (unilateral or bilateral) weighted shift on \(\ell^2\) with \(\| T\|= r(T)\), then \(T\) is reflexive [cf. H. Bercovici, C. Foiaş and C. Pearcy, “Dual algebras with applications to invariant subspaces and dilation theory”, Reg. Conf. Ser. Math. 56 (1985; Zbl 0569.47007), p. 104, Theorem 10.6]. This is more general than Corollary 2.3 here. On the other hand, Theorems 2.5 and 2.6 here are more restrictive than Corollary 2.3 in an earlier paper by the author and Y. N. Dehghan [Southeast Asian Bull. Math. 28, No. 3, 587–593 (2004; Zbl 1081.47039), see the following review].


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
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