Stability of factorization of polynomial operator pencils (Q798941)

The authors assume the existence of a factorization \(P(\lambda)=P_ 1(\lambda)\cdot P_ 2(\lambda)\) for an operator pencil \(C\ni\lambda \to P(\lambda)=\lambda^ n1_ E+\lambda^{n-1}A_ 1+...+A_ n\) where \(A_ 1,...,A_ n\) are bounded operators in a Banach space E. They discuss some properties of factorizations of operator pencils from some neighbourhood of \(P(\lambda)\). Let us assume that \(P(\lambda\),\(\mu)\) depends analytically on a (local) parameter \(\mu \in C\), \(P(\lambda,0)=P(\lambda).\) Some condition for the existence of a factorization \(P(\lambda,\mu)=P_ 1(\lambda,\quad\mu )\cdot P_ 2(\lambda,\mu)\) with \(P_ i(\lambda,0)=P_ i(\lambda), P_ i(\lambda,\mu)\) depends analytically on \(\mu \in C (i=1,2)\), is given.