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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Left definite multiparameter eigenvalue problems

Author: Paul Binding
Journal: Trans. Amer. Math. Soc. 272 (1982), 475-486
MSC: Primary 47A70
MathSciNet review: 662047
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Abstract: We study the problem

$\displaystyle (\ast)\qquad{T_m}{x_m} = \sum\limits_{n = 1}^k {{\lambda _n}{V_{mn}}{x_m},\qquad 0 \ne } {x_m} \in {H_m},\,m = 1, \ldots ,k,$

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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Article copyright: © Copyright 1982 American Mathematical Society