On right inverse operator for matrix operator of generalized convolution (Q1842472)

Let \(H\) be a topological vector space over \(\mathbb{K}\) \((\mathbb{K}\in \{\mathbb{R}, \mathbb{C}\})\), let \((x_\lambda)_{\lambda\in B}\subset H\) be such that \(\overline{\text{spann}\{x_\lambda: \lambda\in B\}}= H\), and let \(Q\) be a family of linear continuous operators \(\ell: H\to H\) such that: (1) \(Q\) is a commutative algebra, (2) for any \(\ell\in Q\) there exists a function \(a: B\to \mathbb{K}\) such that \(\ell(x_\lambda)= a(\lambda) x_\lambda\) for all \(\lambda\in B\), (3) \(\text{id}_H\in Q\). For \(n\geq 2\) consider a matrix operator \(L: H^n\to H^n\), \((Ly)_j= \sum^n_{k= 1} L_{j, k} y_k\), \(y= (y_1, \dots, y_n)\in H^n\), \(j= 1,\dots, n\), where \(L_{j, k}\in Q\), \(j, k= 1,\dots, n\). The author characterizes some cases in which \(L\) has the right inverse in the space of linear operators \(H^n\to H^n\). The results are illustrated by examples with \(H:= H(G)=\) the space of holomorphic functions in a domain \(G\subset \mathbb{C}^p\) and \(x_\lambda:= \exp(\lambda z)\), \(\lambda\in \mathbb{C}^p:= B\).