Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 652444
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

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


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

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,
  • [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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A55, 35P05, 47A10

Retrieve articles in all journals with MSC: 47A55, 35P05, 47A10

Additional Information

Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society