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Imbedding theorems for the invariant subspaces of the backward shift operator
For subspaces K θ p of the form\(K_\theta ^p = H^p \cap \theta \overline {H_o^p } \) of the Hardy space Hp and for measures μ with support in the closed unit circleclos\(\mathbb{D}\), one finds conditions that ensure the imbedding Kθ⊂Lp(μ). One considers measures with support inclos\(\mathbb{D}\), satisfying the following condition: for some number ε>0 and for all circles Δ with center on the circumference, intersecting the set\(\left\{ {z \in \mathbb{D}:\left| {\theta \left( z \right)} \right|< \varepsilon } \right\}\), we have the inequality μ(Δ)⩽Cℓ(Δ). Here C does not depend on Δ, while ℓ(Δ) is the radius of the circle Δ. For such measures one has the imbedding K θ p ⊂Lp(μ). From here one derives a criterion for the imbedding K A 2 ⊂L2(μ), found by B. Cohn for inner functions θ, such that the set\(\left\{ {z \in \mathbb{D}:\left| {\theta \left( z \right)} \right|< \varepsilon } \right\}\) is connected for some positive ɛ. In the paper one also proves that a condition on μ, necessary and sufficient for the imbedding of K θ p into Lp(μ), must depend on p.
قضایای Imbedding برای the نامتغیر با عملگر شیفت عقب
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