The analytic invariant subspace of the n-tuple of commuting operators (Q1103846)

Let X be a complex Banach space, let \(a=(a_ 1,...,a_ n)\) be a commuting n-tuple of bounded linear operators on X, denote by Lat \(a_ i\) the family of all subspaces of X which are invariant under \(a_ i\) and put Lat a\(=\cap^{n}_{i=1}Lat a_ i\); given any open set \(G\subset {\mathbb{C}}^ n\), let \({\mathcal A}(G,X)\) be the space of all X-valued analytic functions on G. The authors introduce the notion of an analytic invariant subspace of a, by which they mean a subspace \(Y\in Lat a\) such that given any polydisc \(D\subset {\mathbb{C}}^ n\), any integer p with \(0\leq p\leq n-1\) and any \(\psi \in \bigwedge^ p[\sigma,{\mathcal A}(D,X)]\) (here \(\sigma =(s_ 1,...,s_ n)\) is an n-tuple of indeterminates, \(\bigwedge^ p\) is the set of all exterior forms of degree p in \(\sigma\), with coefficients in Y) with \(\alpha \psi \in \bigwedge^{p+1}[\sigma,{\mathcal A}(D,Y)]\) \((\alpha (z)=(z_ 1-a_ 1)s_ 1+...+(z_ n-a_ n)s_ n)\), there is \(\Phi \in \bigwedge^ p[\sigma,{\mathcal A}(D,Y)]\), \(\xi \in \bigwedge^{p-1}[\sigma,{\mathcal A}(D,X)]\) such that \(\psi =\Phi +\alpha \xi\). The n-tuple a is said to have the single-value extension property, denoted by \(a\in (A)\), if \(H^ p({\mathcal A}(D,X),\alpha)=0\) for all polydiscs \(D\subset {\mathbb{C}}^ n\) and all integers p with \(0\leq p\leq n-1.\) The authors prove that if \(a\in (A)\) and Y is an analytic invariant subspace of a, then \(a_ y\in (A)\) and \(Sp(a,y)=Sp(a_ y,y)\) for all \(y\in Y\), where \(a_ y=(a_ 1| Y,...,a_ n| Y)\); moreover \(Sp(a,Y)=\cup_{y\in Y}Sp(a,y)\subset Sp(a,X).\) Their other results include a proof that if \(Y\in Lat a\), then Y is an analytic invariant subspace of a if, and only if, \(a^ y\in (A)\); here \(a^ y=(a^ y_ 1,...,a^ y_ n)\), where \(a^ y_ i\) is the operator on X/Y induced by \(a_ i\).