Root functions of boundary eigenvalue operator functions (Q1097486)

Consider the boundary value problem \(f'=-A(\cdot,\lambda)f+g\), \(A(\lambda)f(a)+B(\lambda)f(b)=x\). Here \(\lambda\) is a spectral parameter, A(\(\cdot,\cdot)\) is an \(n\times n\) matrix-valued function, which is continuous on [a,b]\(\times {\mathbb{C}}\) and holomorphic with respect to the second variable on \({\mathbb{C}}\), and A(\(\cdot)\) and B(\(\cdot)\) are holomorphic \(n\times n\) matrix-valued functions on \({\mathbb{C}}\). Let Y(\(\cdot,\lambda)\) be the associated fundamental matrix. The authors describe the relationships between the root functions of the operator function \(T(\lambda)f=(f'+A(\cdot,\lambda)f\), \(A(\lambda)f(a)+B(\lambda)f(b))\), \(f\in H_ 1(a,b))^ n\), and those of the characteristic matrix function \(M(\lambda)=A(\lambda)Y(a,\lambda)+B(\lambda)Y(b,\lambda)\). The main result states a method to construct a biorthogonal canonical systems of root functions of T and \(T^*\) if biorthogonal canonical systems of root functions of M and \(M^*\) are given. The latter result is first proved in an abstract framework and next applied to concrete classes of boundary value problems. In a subsequent paper [Integral Equations and Operator Theory 9, 275-285 (1986; Zbl 0589.47014)] the reviewer has shown that the operator function T is analytically equivalent on \({\mathbb{C}}\) to a simple extension of the matrix function M. This equivalence yields the above mentioned connection between the root functions of the two functions as a corollary.