A note on a completeness result for a quadratic polynomial operator (Q1196591)

This paper presents the completeness theorem for eigenvectors of a quadratic operator polynomial \(D(\lambda)=\lambda^{2l}-\lambda A_ 1- A_ 2\), where \(A_ 1\) and \(A_ 2\) are compact selfadjoint operators and \(A_ 2\) is assumed to be positive definite. A connection with the boundary value problem of a vibrating membrane is discussed. [Note from the reviewer: other and more general results in this direction may be found in the paper by \textit{A. A. Shkalikov}, Math. USSR, Sb. 63, No. 1, 97-119 (1989; Zbl 0668.47010)].