Abstract
For an integer p ≥ 1, let Γp be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm NΓp(.) and the property
$$\begin{array}{}
\displaystyle
\sum_{k=1}^{\infty} |\lambda_k(A)|^p\le a_p N_{{\it\Gamma}_p}^p(A)
\;\;(A\in {\it\Gamma}_p),
\end{array}$$
where λk(A) (k = 1, 2, …) are the eigenvalues of A and ap is a constant independent of A. Let A, Ã ∈ Γp and
$$\begin{array}{}
\displaystyle
{\it\Delta}_p(A, \tilde A):= N_{{\it\Gamma}_p}(A-\tilde A)
\;\exp\;\left[a_p b_p^p \;\left(1+\frac{1}2
(N_{{\it\Gamma}_p}(A+\tilde A) + N_{{\it\Gamma}_p}(A-\tilde A))\right)^p\right],
\end{array}$$
where bp is the quasi-triangle constant in Γp. It is proved the following result: let I be the unit operator, I – Ap be boundedly invertible and
$$\begin{array}{}
\displaystyle
{\it\Delta}_p(A, \tilde A)\exp\;\left[\frac{a_pN^p_{{\it\Gamma}_p}(A) }
{\psi_p(A)}\right] \lt 1,
\end{array}$$
where ψp(A) = infk=1,2,… |1 –
$\begin{array}{}
\displaystyle
\lambda_k^{p}
\end{array}$(A)|. Then I – Ãp is also boundedly invertible. Applications of that result to the spectrum perturbations of absolutely p-summing and absolutely (p, 2) summing operators are also discussed. As examples we consider the Hille-Tamarkin integral operators and matrices.
The distinguished boundary of the unit operator ball
