Basis problems for matrix valued functions

Abstract

This article deals with spectral and basis properties of matrix valued functions \(L(\lambda): [a,b] \to S({\mathbb C}^n)\), where \(S({\mathbb C}^n)\) is the space of selfadjoint matrices from \(M({\mathbb C})^n\). The main results are as follows. If \(L\) is continuously differentiable and the equation \((L(\lambda)x,x) = 0\) has a unique solution \(p(x) \in [a,b]\) for all \(x \neq 0\) with \((L'(p(x)x,x) > 0\), then the eigenvectors of \(L\) corresponding to eigenvalues from \([a,b]\) form a basis in \({\mathbb C}^n\). If \(L\) is only continuous and the equation \((L(\lambda)x,x) = 0\) has a unique solution \(p(x) \in [a,b]\) for all \(x \neq 0\) with increasing (at \(\lambda = p(x)\)) \((L(\lambda)x,x)\), then the eigenvectors of \(L\) corresponding to eigenvalues from \([a,b]\) also form a basis in \({\mathbb C}^n\). At the end of the article, the author makes some comments on the infinite-dimensional variants of these problems.