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Multipliers of weighted \(l^ p\) spaces. (English) Zbl 0747.47021

Let \(\omega\) be a positive sequence defined on the non-negative integers \(\mathbb{Z}^ +\) and, for \(1\leq p<\infty\), let \(\ell^ p(\omega)\) denote the space of complex sequences \(\{x(n)\}_{n\in\mathbb{Z}^ +}\) such that \(\sum^ \infty_{n=0}| x(n)|^ p\omega(n)^ p<\infty\) with the natural norm. Suppose further that \(\omega\) is normalized so that \(\omega(0)=1\) and that, for each \(k\in\mathbb{Z}^ +\), \(\sup_{n\geq 0}\omega(n+k)/\omega(n)<\infty\), so that the unilateral shift is bounded on \(\ell^ p(\omega)\). Let \(m^ p(\omega)\) be the space of complex sequences \(\alpha\) with the property that \(\alpha*x\in\ell^ p(\omega)\) whenever \(x\in\ell^ p(\omega)\) and let \({\mathcal M}^ p(\omega)\) denote the space of (necessarily bounded) linear operators on \(\ell^ p(\omega)\) of the form \(T_ \alpha:x\to\alpha*x\) for \(\alpha\in{\mathcal M}^ p(\omega)\). The elements of \({\mathcal M}^ p(\omega)\) are called the multipliers of \(\ell^ p(\omega)\). Endow \(m^ p(\omega)\) with the norm inherited from the operator norm on \({\mathcal M}^ p(\omega)\).
It is readily seen that, if \(\ell^ p(\omega)\) is an algebra under convolution, then \(m^ p(\omega)=\ell^ p(\omega)\) with equivalence of norms, and hence \({\mathcal M}^ p(\omega)={\mathcal A}^ p(\omega)\), where \({\mathcal A}^ p(\omega)\) is the closure in \({\mathcal M}^ p(\omega)\) with respect to the operator norm of the space of multipliers \(T_ \alpha\) associated with elements \(\alpha\) in \(m^ p(\omega)\) having finite support. The main aim of the paper is to prove a partial converse to this result as follows. It is shown that, if \({\mathcal M}^ p(\omega)={\mathcal A}^ p(\omega)\) and the weight \(\omega\) is regulated, then \(\ell^ p(\omega)\) is an algebra under convolution. To say that \(\omega\) is regulated means that there exists a positive integer \(k\) such that \(\omega(n+k)/\omega(n)\to 0\) as \(n\to\infty\). This converse result provides a partial answer to a question raised by A. L. Shields [Topics in Operator Theory, Math. Surveys 13, 49-128 (1974; Zbl 0303.47021)] for the case \(p=2\) and is related to a result of W. G. Bade, H. G. Dales and K. B. Laursen [Multipliers of radical Banach algebras of power series, Mem. Amer. Math. Soc. 303 (1984; Zbl 0538.43002)] for the case \(p=1\).
The paper also gives a description of the compact multipliers as those multipliers \(T_ \alpha\) in \({\mathcal A}^ p(\omega)\) for which \(\alpha(k)=0\) whenever \(k<k_ \omega\), where \(k_ \omega\) is the smallest integer occurring in the definition of regularity for \(\alpha\), with the convention that \(k_ \omega=\infty\) if \(\alpha\) is not regulated.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
46J05 General theory of commutative topological algebras
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