Multiparameter Variational Principles

Abstract

Let T and V be self-adjoint operators on a separable Hilbert space H, where V is bounded and T has compact resolvent. Then a spectral theory, including eigenvector completeness, may be given variationally for the eigenvalue problem \[ Tx = \lambda Vx,\quad 0 \ne x \in H\] provided either $T \gg 0$ (i.e., $(x,Tx) \geq \alpha \| x \|^2 $ for some $\alpha > 0$) or $V \gg 0$. These conditions are known as left and right definiteness (LD and RD) respectively. Generalisations of LD and RD appropriate for the multiparameter problem \[ T_m x_m = \sum_{n = 1}^k {\lambda _n } V_{mn} x_m ,\quad 0 \ne x_m \in H_{m} ,\quad m = 1, \cdots ,k,\] have already been established in the literature. New variational principles are given here in terms of $\mathbb{R}^k $-valued functions on $ \times _{m = 1}^k H_m $ and $ \otimes _{m = 1}^k H_m $. In particular, when LD is assumed, a variational spectral theory is given, including eigenvector completeness. Such a theory is shown to be impossible under RD, although a finite dimensional version is established for a condition including both LD and RD.