Perturbations preserving asymptotics of spectrum with remainder

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.