Best Approximations by Increasing Invariant Subspaces of Self-Adjoint Operators

The paper describes approximations properties of monotonically increasing sequences of invariant subspaces of a self-adjoint operator, as well as their symmetric generalizations in a complex Hilbert space, generated by its positive powers. It is established that the operator keeps its spectrum over the dense union of these subspaces, equipped with quasi-norms, and that it is contractive. The main result is an inequality that provides an accurate estimate of errors for the best approximations in Hilbert spaces by these invariant subspaces.