Backward shift invariant subspaces in the bidisc II

For every invariant subspace NI in the Hardy spaces H2 (f2 ), let Vz and Vw be mulitplication operators on AL Then it is known that the condition Vz V; v;vz on NI holds if and only if J;I is a Demling type invariant subspace. For a backward shift invariant subspace N in H2(f2), two operators Sz and Sw on N are defined by Sz = PN LzPN and Sw = PN Lw PN, where PN is the orthogonal projection from L2(f2) onto N. It is given a characterization of N satisfying szs1:J = s1:JsZ on N

(A_2)-Conditions and Carleson Inequalities

Let v and µ be finite positive measures on the open unit disk D. We say that v and µ satisfy the ( v ,µ )-Carleson inequality, if there is a constant C > 0 such that 111 dv C D 111 dµ for all analytic polynomials 1 . In this paper, we study the necessary and sufficient condition for the ( v , µ )-Carleson inequality. We establish it when v or µ is an absolutely continuous measure with respect to the Lebesgue area measure which satisfies the (A )-condition. Moreover, many concrete examples of such measures are given

Riesz's Functions In Weighted Hardy And Bergman Spaces

Let µ be a finite positive Borel measure on the closed unit disc D. For each a in D, put S(a) = inf k If IP dµ where f ranges over all analytic polynomials with f(a) = 1. This upper semicontinu­ous function S(a) is called a Riesz's function and studied in detail. Moreover several applications are given to weighted Bergman and Hardy spaces

Backward shift invariant subspaces in the bidisc II

For every invariant subspace NI in the Hardy spaces H2 (f2 ), let Vz and Vw be mulitplication operators on AL Then it is known that the condition Vz V; v;vz on NI holds if and only if J;I is a Demling type invariant subspace. For a backward shift invariant subspace N in H2(f2), two operators Sz and Sw on N are defined by Sz = PN LzPN and Sw = PN Lw PN, where PN is the orthogonal projection from L2(f2) onto N. It is given a characterization of N satisfying szs1:J = s1:JsZ on N