TY - JOUR

T1 - On embedding theorems for coinvariant subspaces of the shift operator. II

AU - Александров, Алексей Борисович

N1 - Funding Information: This research was supported in part by the Russian Foundation for Basic Research, grant 99-01-00103. Translated by A. B. Aleksandrov.

PY - 2002

Y1 - 2002

N2 - For every inner function Θ, we set Θ*(H2) =def H2⊖ΘH2 and Θ*(HP) =def closHP(HP ∩ Θ*(Hp)) if p ≠ 2. Denote by C P(Θ) the set of all (complex) measures μ, on the closed unit disk double-struck D̄ such that Θ*(Hp) ⊂ L P(|μ|). An inner function Θ is said to be one-component if the set (z ∈ double-struck D:|Θ(z)| < ε) is connected for some ε ∈ (0, 1). A series of criteria for an inner function to be one-component are obtained. For example, it is proved that Θ is one-component if and only if Cp(Θ) does not depend on p ∈ (0,+∞). Moreover, a criterion in terms of reproducing kernels of Θ *(H2) for an inner function Θ to be one-component is obtained. The set CP(Θ) is described in the case where Θ is a Blaschke product of special form. This description implies that the set of all p such that a given measure μ belongs to Cp(Θ) can have any finite or infinite number of connected components. The following examples of interpolating Blaschke products Θ and positive measures μ are constructed: (1) Θ*(H1) ⊂ L1(μ) and Θ*(H2) ⊂ L2(μ) but Θ* (Hp) ⊄ LP(μ) for any p ∈ (1, 2); (2) Θ*(HP) ⊂ Lp(n) if and only if p = 1/n, where n is a positive integer; (3) Θ*(HP) ⊂ L p(μ) if and only if p ≠ 1/n, where n is a positive integer.

AB - For every inner function Θ, we set Θ*(H2) =def H2⊖ΘH2 and Θ*(HP) =def closHP(HP ∩ Θ*(Hp)) if p ≠ 2. Denote by C P(Θ) the set of all (complex) measures μ, on the closed unit disk double-struck D̄ such that Θ*(Hp) ⊂ L P(|μ|). An inner function Θ is said to be one-component if the set (z ∈ double-struck D:|Θ(z)| < ε) is connected for some ε ∈ (0, 1). A series of criteria for an inner function to be one-component are obtained. For example, it is proved that Θ is one-component if and only if Cp(Θ) does not depend on p ∈ (0,+∞). Moreover, a criterion in terms of reproducing kernels of Θ *(H2) for an inner function Θ to be one-component is obtained. The set CP(Θ) is described in the case where Θ is a Blaschke product of special form. This description implies that the set of all p such that a given measure μ belongs to Cp(Θ) can have any finite or infinite number of connected components. The following examples of interpolating Blaschke products Θ and positive measures μ are constructed: (1) Θ*(H1) ⊂ L1(μ) and Θ*(H2) ⊂ L2(μ) but Θ* (Hp) ⊄ LP(μ) for any p ∈ (1, 2); (2) Θ*(HP) ⊂ Lp(n) if and only if p = 1/n, where n is a positive integer; (3) Θ*(HP) ⊂ L p(μ) if and only if p ≠ 1/n, where n is a positive integer.

UR - http://www.scopus.com/inward/record.url?scp=2342523549&partnerID=8YFLogxK

U2 - 10.1023/A:1015379002290

DO - 10.1023/A:1015379002290

M3 - Article

AN - SCOPUS:2342523549

VL - 110

SP - 2907

EP - 2929

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 5

ER -