The spectral theory of Toeplitz operators applied to approximation problems in Hilbert spaces (Q814799)

The paper is devoted to investigation of a so called constrained approximation problem. Let a Hilbert space \(H\) be decomposed in two different ways: \[ H=H_+\oplus H_-=K_+\oplus K_- . \] In the examples considered, \(H_+\) is a subspace of analytic functions (Hardy, Segal-Bergman) and \(K_+\) is a subspace of functions with restricted support. The approximation problem under the consideration is to approximate a given \(k_1\in K_+\), as well as possible, by the projection onto \(K_+\) of a member of \(H_+\). The corresponding extremal problem is considered. This problem is studied separately when there holds a certain denseness condition: \(P_{K_+}H_+\) is dense in \(K_+\), or equivalently, \(K_+\cap H_-=\{0\}\), and when this condition does not hold. Constrained approximation in the Hardy spaces on an annulus and a strip is specially treated.