On local spectral properties of operator matrices

In this paper, we focus on a $2 \times 2$ 2 × 2 operator matrix $T_{\epsilon _{k}}$ T ϵ k as follows: $$\begin{aligned} T_{\epsilon _{k}}= \begin{pmatrix} A & C \\ \epsilon _{k} D & B\end{pmatrix}, \end{aligned}$$ T ϵ k = ( A C ϵ k D B ) , where $\epsilon _{k}$ ϵ k is a positive sequence such that $\lim_{k\rightarrow \infty }\epsilon _{k}=0$ lim k → ∞ ϵ k = 0 . We first explore how $T_{\epsilon _{k}}$ T ϵ k has several local spectral properties such as the single-valued extension property, the property $(\beta )$ ( β ) , and decomposable. We next study the relationship between some spectra of $T_{\epsilon _{k}}$ T ϵ k and spectra of its diagonal entries, and find some hypotheses by which $T_{\epsilon _{k}}$ T ϵ k satisfies Weyl’s theorem and a-Weyl’s theorem. Finally, we give some conditions that such an operator matrix $T_{\epsilon _{k}}$ T ϵ k has a nontrivial hyperinvariant subspace.

Krylov improvements of the Uzawa method for Stokes type operator matrices

The paper is devoted to Krylov type modifications of the Uzawa method on the operator level for the Stokes problem in order to accelerate convergence. First block preconditioners and their effect on convergence are studied. Then it is shown that a Krylov–Uzawa iteration produces superlinear convergence on smooth domains, and estimation is given on its speed.