Left definite multiparameter eigenvalue problems

Binding, Paul

doi:10.1090/S0002-9947-1982-0662047-3

Left definite multiparameter eigenvalue problems
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Trans. Amer. Math. Soc. 272 (1982), 475-486 Request permission

Abstract:

We study the problem \[ (\ast )\qquad {T_m}{x_m} = \sum \limits _{n = 1}^k {{\lambda _n}{V_{mn}}{x_m},\qquad 0 \ne } {x_m} \in {H_m}, m = 1, \ldots ,k,\] where ${T_m}$ and ${V_{mn}}$ are selfadjoint linear operators on separable Hilbert spaces ${H_m}$, with ${T_m}$ positive, $T_m^{ - 1}$ compact and ${V_{mn}}$ bounded. We assume “left definiteness” which involves positivity of certain linear combinations of cofactors in the determinant with $(m, n)$th entry $({x_m}, {V_{mn}}{x_m})$. We establish a spectral theory for $(\ast )$ that is in some way simpler and more complete than those hitherto available for this case. In particular, we make use of operators ${B_n} = \Delta _n^{ - 1}{\Delta _0}$, where the ${\Delta _n}$ are determinantal operators on $\otimes _{m = 1}^k{H_m}$. This complements an established approach to the alternative “right definite” problem (where ${\Delta _0}$ is positive) via the operators ${\Gamma _n} = \Delta _0^{ - 1}{\Delta _n}$.

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