Variational Principles for Eigenvalues of Compact Operators
Giles Auchmuty · SIAM Journal on Mathematical Analysis · 1989
Variational principles for finding real and complex nonzero eigenvalues, and associated eigenvectors, of a linear compact operator K on a Hilbert space are developed and analyzed. When K is self-adjoint, certain unconstrained variational problems are described for finding the positive, respectively, negative, eigenvalues of K and the corresponding eigenvectors. These principles are extended to generalized eigenproblems and to nonlinear compact operators. For nonself-adjoint linear operators, a minimization problem for certain positive real eigenvalues is described. All the positive real eigenvalues may be described as critical points of a Lagrangian functional. These characterizations are then extended to describe complex eigenvalues and eigenvectors of nonself-adjoint, compact linear operators.