Yousefi, B. Strictly cyclic algebra of operators acting on Banach spaces \(H^p(\beta)\). (English) Zbl 1049.47033 Czech. Math. J. 54, No. 1, 261-266 (2004). Summary: Let \(\{\beta (n)\}^{\infty }_{n=0}\) be a sequence of positive numbers and \(1 \leq p < \infty \). We consider the space \(H^{p}(\beta )\) of all power series \(f(z)=\sum ^{\infty }_{n=0}\hat {f}(n)z^{n}\) such that \(\sum ^{\infty }_{n=0}| \hat {f}(n)| ^{p}\beta (n)^{p} < \infty \). We investigate strict cyclicity of \(H^{\infty }_{p}(\beta)\), the weakly closed algebra generated by the operator of multiplication by \(z\) acting on \(H^{p}(\beta)\), and determine the maximal ideal space, the dual space and the reflexivity of the algebra \(H^{\infty }_{p}(\beta)\). We also give a necessary condition for a composition operator to be bounded on \(H^{p}(\beta)\) when \(H^{\infty }_{p}(\beta)\) is strictly cyclic. Cited in 7 Documents MSC: 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A25 Spectral sets of linear operators Keywords:Banach space of formal power series associated with a sequence; bounded point evaluation; strictly cyclic maximal ideal space; Schatten \(p\)-class; reflexive algebra; semisimple algebra; composition operator PDFBibTeX XMLCite \textit{B. Yousefi}, Czech. Math. J. 54, No. 1, 261--266 (2004; Zbl 1049.47033) Full Text: DOI EuDML References: [1] J. B. Conway: A Course in Functional Analysis. Springer-Verlag, New York, 1985. · Zbl 0558.46001 [2] J. B. Conway: The Theory of Subnormal Operators. American Mathematical Society, 1991. · Zbl 0743.47012 [3] K. Seddighi, K. Hedayatiyan and B. Yousefi: Operators acting on certain Banach spaces of analytic functions. Internat. J. Math. Sci. 18 (1995), 107-110. · Zbl 0821.47022 · doi:10.1155/S0161171295000147 [4] A. L. Shields: Weighted shift operators and analytic function theory. Math. Survey, A.M.S. Providence 13 (1974), 49-128. · Zbl 0303.47021 [5] B. Yousefi: On the space \(\ell ^{p}(\beta )\). Rend. Circ. Mat. Palermo Serie II XLIX (2000), 115-120. · Zbl 0952.47027 · doi:10.1007/BF02904223 [6] B. Yousefi: Bounded analytic structure of the Banach space of formal power series. Rend. Circ. Mat. Palermo Serie II LI (2002), 403-410. · Zbl 1194.47035 · doi:10.1007/BF02871850 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.