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Proceedings of the American Mathematical Society

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Perturbations preserving asymptotics of spectrum with remainder


Author: A. G. Ramm
Journal: Proc. Amer. Math. Soc. 85 (1982), 209-212
MSC: Primary 47A55; Secondary 35P05, 47A10
DOI: https://doi.org/10.1090/S0002-9939-1982-0652444-X
MathSciNet review: 652444
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Abstract: Let $ A$ be a compact linear operator on a Hilbert space $ H$, $ {s_n}(A) = \lambda _n^{1/2}({A^* }A)$, $ Q$ be a linear operator, $ \vert Qf\vert \leqslant c\vert Af{\vert^a}\vert f{\vert^{1 - a}}$, $ a > 0$, $ c > 0$, $ \forall f \in H$. Let

$\displaystyle {s_n}(A) = c{n^{ - r}}\{ 1 + O({n^{ - q}})\} ,\quad r,q > 0,B = A(I + Q).$

Then

$\displaystyle {s_n}(B) = {s_n}(A)\{ 1 + O({n^{ - \gamma }})\} ,\quad \gamma = \min \left( {q,\frac{{ra}} {{1 + ra}}} \right).$

Some applications of this result to the spectral theory of elliptic operators are given.

References [Enhancements On Off] (What's this?)

  • [1] A. G. Ramm, Perturbations preserving asymptotic of spectrum, J. Math. Anal. Appl. 76 (1980), no. 1, 10–17. MR 586640, https://doi.org/10.1016/0022-247X(80)90055-4
  • [2] Alexander G. Ramm, Theory and applications of some new classes of integral equations, Springer-Verlag, New York-Berlin, 1980. MR 601947
  • [3] S. G. Kreĭn, Linear differential equations in Banach space, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin; Translations of Mathematical Monographs, Vol. 29. MR 0342804
  • [4] Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci., U. S. A. 37 (1951), 760–766. MR 0045952

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0652444-X
Article copyright: © Copyright 1982 American Mathematical Society