engleski

On relative perturbation theory for block operator matrices

We use weakly-formulated operator equations to study perturbation problems for 22 operator block matrices. We allow (integro) differential operators, defined by quadratic forms, as coefficients in these weak operator equations and block operator matrices. An analysis of the ``weakly formulated'' Sylvester equation yields new scaling robust bounds for the rotation of spectral subspaces of a nonnegative self-adjoint operator in a Hilbert space. Our bound extends the known results of Davis and Kahan. As an example we give constructive estimates for the convergence rates of eigenvalues and eigenfunctions of the Arch model eigenvalue problem as the diameter of the thin elastic (rod like) body tends to zero. Furthermore, this prototype problem is identified as representative for the whole class of non-inhibited stiff perturbation families. This class of perturbations is characterized by a kind of inf-sup condition. Another problem which we study is bounding a perturbation of the square root of a positive self-adjoint operator. The obtained estimate is also of the relative type and we use a Sylvester like equation to solve this problem (cf. Grubišić, L. ; Veselić, K. On weakly formulated Sylvester equations and applications, Integral Equations and Operator Theory, to appear, Preprint: http://arxiv.org/abs/math.SP/0507532). This is a joint work with K. Veselić, Hagen, Germany and J. Tambača, Zagreb, Croatia.

theory of perturbations; operator matrices

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