Equivalence of boundary eigenvalue operator functions and their characteristic matrix functions
Let \(E,G,F_ 1,F_ 2\) be Banach spaces and \(T^ D\in L(E,F_ 1)\), \(T^ R\in L(E,F_ 2)\) and \(Z\in L(G,E)\). Set \(F:=F_ 1\times F_ 2\) and \(T:=(T^ D,T^ R)\). Assume that \(T^ D\) is right invertible, Z is injective and \(N(T^ D)=R(Z)\). Then T is equivalent to a simple extension of its ''characteristic'' matrix \(M:=T^ RZ\). Explicitly, \(T=C\left( \begin{matrix} M\\ 0\end{matrix} \begin{matrix} 0\\ I_{F_ 1}\end{matrix} \right)D\), where \(C=\left( \begin{matrix} 0\\ I_{F_ 2}\end{matrix} \begin{matrix} I_{F_ 1}\\ T^ RL\end{matrix} \right)\), \(D=(Z,L)^{-1}\) and L is a right inverse of \(T^ D\). If T depends analytically on a parameter \(\lambda\) which varies in an open set \(\Omega\) \(\subset {\mathbb{C}}\), then T is called an ''abstract boundary eigenvalue operator function'' and the above stated equivalence is globally analytic on \(\Omega\). This analytic equivalence yields explicit formulas for canonical systems of root functions of T in terms of such systems of the ''matrix'' function M. Applications to boundary eigenvalue problems for n-th order systems of differential equations are given. In these applications \(T^ D\) is defined by differential operators and \(T^ R\) by boundary conditions; M is a characteristic matrix corresponding to the boundary value problem \(Ty=0\). In the special case that \(T^ D\) is a first order differential operator, the equivalence stated above was proved by \textit{M. A. Kaashoek} [Integral Equations Oper. Theory 9, 275-285 (1986; Zbl 0589.47014)].
abstract boundary eigenvalue operator function
scientific article; zbMATH DE number 3992220