This monograph contains an exposition of the foundations of the
spectral theory of polynomial operator pencils acting in a Hilbert space.
Spectral problems for polynomial pencils have attracted a steady interest in
the last 35 years, mainly because they arise naturally in such diverse areas
of mathematical physics as differential equations and boundary value problems,
controllable systems, the theory of oscillations and waves, elasticity theory,
and hydromechanics.
In this book, the author devotes most of his attention to the fundamental
results of Keldysh on multiple completeness of the eigenvectors and associate
vectors of a pencil, and on the asymptotic behavior of its eigenvalues and
generalizations of these results. The author also presents various theorems on
spectral factorization of pencils which grew out of known results of M. G.
Kreibreven and Heinz Langer. A large portion of the book involves the theory
of selfadjoint pencils, an area having numerous applications. Intended for
mathematicians, researchers in mechanics, and theoretical physicists interested
in spectral theory and its applications, the book assumes a familiarity with
the fundamentals of spectral theory of operators acting in a Hilbert space.
