Inequalities for eigenvalues of selfadjoint operators

Author:
Stephen M. Hook

Journal:
Trans. Amer. Math. Soc. **318** (1990), 237-259

MSC:
Primary 47A70; Secondary 35P05, 47B25, 49G20

MathSciNet review:
943604

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Abstract: We establish several inequalities for eigenvalues of selfadjoint operators in Hilbert space. The results are quite general. In particular, let be a region in its boundary and the Laplace operator in . Let be a polynomial of degree having nonnegative real coefficients. We show that if the problems

(1) in on ;

(2) in and its first derivatives on; and

(3) in and its first derivatives on are selfadjoint with discrete spectra of finite multiplicity etc. then

(4) for each index . The set of problems (1), (2), (3) and the result (4) is only one example of our more general result.

The above problems (1), (2), and (3) can be thought of as related through the single operator given by the Laplacian. We also establish results for eigenvalues for unrelated operators. Let , and be selfadjoint on domains , and with . If , , and have discrete spectra and arranged in ascending order, as above, then inequality

(5) is established for each positive integer .

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1990-0943604-8

Article copyright:
© Copyright 1990
American Mathematical Society