Commutants of certain operators in the space of analytic functions of several variables (Q1084316)

This paper considers spaces of functions of n complex variables, which are analytic in the polydisc \({\mathcal D}_ R\) in \(C^ n\), where \(R=(R_ 1,...,R_ n)\), \(0<R_ j<\infty\), and \({\mathcal D}_ R=\{z=(z_ 1,...,z_ n)\in C^ n:| z_ j| <R_ j\), \(j=1,...,n\}\). The results in this paper generalize known results in the single variable case. \(A_ R\) is the space of all functions analytic in \({\mathcal D}_ R\), with the topology of uniform convergence on compact subsets of \({\mathcal D}_ R\); this topology can be defined in terms of certain projective limits. \(\bar A{}_ R\) is a space of functions analytic in the closure of \({\mathcal D}_ R\), with topology defined analogously in terms of inductive limits. On these spaces, let \(U_ j\) represent multiplication by \(z_ j\) \((j=1,...,n)\), and let \(\Delta_ j\) represent ''division by \(z_ j''\) \((j=1,...,n):\) \[ (\Delta_ jf)(z)=[f(z_ 1,...,z_ j,...,z_ n)-f(z_ 1,...,0,...,z_ n)]/z_ j. \] Consider the problem of describing, for given positive integers \(p_ 1,...,p_ n\), those continuous operators T on \(A_ R\) (or on \(\bar A{}_ R)\) which commute with all the operators \(U_ j^{p_ j}\), \(j=1,...,n\). The general form of such an operator T is given, and a necessary and sufficient condition is derived, in terms of the general form of T, for T to be an isomorphism. This, in turn, gives rise to a condition for a collection of functions to be a quasipower basis, i.e., a basis of the form \(\{Tz^ k\}\) for some isomorphism T, where \(z^ k\) represents \(z_ 1^{k_ 1}...z_ n^{k_ n}\), for nonnegative integers \(k_ 1,...,k_ n\). Also given is a condition that a function f be strongly cyclic with respect to \(U_ 1^{p_ 1},...,U_ n^{p_ n}\), in the sense that, for each function g there exists an operator T commuting with every \(U_ j^{p_ j}\), \(j=1,...,n\), such that \(Tf=g.\) In the special case \(p_ 1=...=p_ n=1\), T is just multiplication by a function \(\phi\), and T can be represented as a power series in the operators \(U_ 1,...,U_ n.\) Results similar to those above are also obtained for the operators \(\Delta_ 1,...,\Delta_ n\), with the exception that it is shown that no function f is strongly cyclic with respect to \(\Delta_ 1^{p_ 1},...,\Delta_ n^{p_ n}\).