engleski

Numerical linear algebra for spectral theory of block operator matrices

We present a perturbation theory for sign-indefinite quadratic forms in a Hilbert space. Under an additional qualitative assumption on the structure of the form, which is in analogy to the structure of the so called quasi-definite matrices from Linear Algebra, we prove an operator representation theorem. Special emphasis has been set on analysis of quadratic forms which are unbounded at both ends and which can be tackled by our algebraic theory in a natural way. Furthermore, with the help of weakly formulated Riccati equations we obtain subspace perturbation theorems for these “quasi-definite operators” and present accompanying estimates on the perturbation of the spectra. Our estimates for the rotation of invariant subspaces have a form of a “relative” tan 2 theorem and are a new result even in the standard matrix case. The example of the Stokes block matrix operator—which is associated to the Cosserat eigenvalue problem—is used to illustrate the theory and to show that our estimates can be attained. As another application we present estimates for the strong solution of the Stokes system in a Lipschitz domain and favorably compare these results with considerations from [1, Section 6 and 7]. The title of the presentation has been motivated by the paper [5], which uses linear algebra in the spectral theory in another context.

perturbation theory; systems of differential equations

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