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A local algebra structure for \(H^ p\) of the polydisc. (English) Zbl 0766.46040

N. M. Wigley [Can. Math. Bull. 18, 597-603 (1975; Zbl 0324.46051)] showed that, for \(p\geq 1\), \(H^ p(\Delta)\) can be given a Banach algebra structure under the Duhamel product given by \[ f*g(z)={d\over dz}\int^ z_ 0f(z-t)g(t)dt, \] where \(\Delta\) is the open unit disc. The authors define on \(H^ p(\Delta^ n)\) a natural extension of the Dunhamel product as follows: \[ f*g(z_ 1,\dots,z_ n)={\partial^ n\over\partial z_ 1,\dots,\partial z_ n}\int^{z_ n}_ 0\cdots\int^{z_ 1}_ 0f(z_ 1-t_ 1,\dots,z_ n-t_ n)g(t_ 1,\dots,t_ n)dt_ 1, \cdots,dt_ n. \] And they show that \(H^ p(\Delta^ n)\), \(0<p\leq\infty\), is a local (Banach or \(F\)-) algebra. This result follows from a more general study on vector-valued analytic functions: For \(p\geq 1\) let \(B\) be a Banach algebra and for \(0<p<1\) let \(B\) be a \(p\)-normed \(F\)-algebra. Let \(H^ p(\Delta,B)\) be the space of \(B\)-valued analytic functions for which the usual \(H^ p\) norm is finite. The authors define the Duhamel product on \(H^ p(\Delta,B)\) and show that \(H^ p(\Delta,B)\) is a Banach algebra for \(p\geq 1\) and an \(F\)- algebra for \(0<p<1\). They also show that there is a natural isomorphism between the maximal ideal space of \(H^ p(\Delta,B)\) and that of \(B\).
Reviewer: J.Wada (Tokyo)

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables

Citations:

Zbl 0324.46051
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