Lectures on the shift operator. II

N. K. Nikol'skii

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part V, Tome 47 (1974), pp. 90-119

Résumé

The topics considered are: bases of eigenfunctions and interpolation; the general theorem of Bari on bases; the biorthogonality of eigenfunction families of operators $PS|K$ and $S^{*}|K$; the uniform minimality; spectral projections; interpolation problems; the test, of Schur; the Carleson condition; the theorem of Shapiro and Shields on $H^2$-interpolation; exponential bases and Muntz's theorem; the inversion of the von Neumann inequality; the interpolation in $H^{\infty}$; the commutant: Lorch–Grinblum theorem on bases, the description of operators commuting with a projection of the shift, a deduction of interpolation theorems of Carleson, Caratheodory–Fejer, Nevanlinna–Pick and other corollaries. The addendum contains a brief surfey and a new proof of the Carleson embedding lemma due to S. A. Vinogradov.