Asymptotic bounds for the eigenvalues and eigenvectors of a perturbed linear non-self-conjugate operator

We obtain asymptotic bounds for the perturbed eigenvalues and eigenvectors of a perturbed linear bounded operator $A(\varepsilon)$, in a Hilbert space under the assumption that $A(\varepsilon)$ is holomorphic at the point $\varepsilon=\varepsilon_0$ and the eigenvalue $\lambda_0=\lambda(\varepsilon_0)$ of the operator $A(\varepsilon_0)$ is isolated and of finite multiplicity. We study certain cases of high degeneracy in the limiting problem, i.e., the case when there are generalized associated vectors.