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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Left definite multiparameter eigenvalue problems
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by Paul Binding PDF
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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 272 (1982), 475-486
  • MSC: Primary 47A70
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0662047-3
  • MathSciNet review: 662047