On an Expansion Theorem of F. V. Atkinson and P. Binding
Hans Volkmer · SIAM Journal on Mathematical Analysis · 1987
We study an expansion theorem of F. V. Atkinson and P. Binding for the right definite multiparameter eigenvalue problem \[ x_m = T_m x_m + \sum\limits_{n = 1}^k {\lambda _n V_{mn} x_m ,\quad m = 1, \cdots ,k} \] where $\lambda _1 , \cdots ,\lambda _k $ are real, $x_m $ is a nonzero element of some separable Hilbert space and $T_m $, $V_{mn} $ are compact symmetric. We present a new version of the theorem which is easier to use in order to prove completeness of eigenvectors under some additional assumptions. In particular, we prove completeness of eigenvectors (i) under Minkowski’s definiteness condition and (ii) for arbitrary two parameter problems.