Perturbations preserving asymptotics of spectrum with remainder
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- by A. G. Ramm PDF
- Proc. Amer. Math. Soc. 85 (1982), 209-212 Request permission
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, $|Qf| \leqslant c|Af{|^a}|f{|^{1 - a}}$, $a > 0$, $c > 0$, $\forall f \in H$. Let \[ {s_n}(A) = c{n^{ - r}}\{ 1 + O({n^{ - q}})\} ,\quad r,q > 0,B = A(I + Q).\] Then \[ {s_n}(B) = {s_n}(A)\{ 1 + O({n^{ - \gamma }})\} ,\quad \gamma = \min \left ( {q,\frac {{ra}} {{1 + ra}}} \right ).\] Some applications of this result to the spectral theory of elliptic operators are given.References
- A. G. Ramm, Perturbations preserving asymptotic of spectrum, J. Math. Anal. Appl. 76 (1980), no. 1, 10–17. MR 586640, DOI 10.1016/0022-247X(80)90055-4
- Alexander G. Ramm, Theory and applications of some new classes of integral equations, Springer-Verlag, New York-Berlin, 1980. MR 601947
- S. G. Kreĭn, Linear differential equations in Banach space, Translations of Mathematical Monographs, Vol. 29, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin. MR 0342804
- 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 45952, DOI 10.1073/pnas.37.11.760
Additional Information
- © Copyright 1982 American Mathematical Society
- 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