Trace ideal criteria for embeddings and composition operators on model spaces

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• 作者：Alexandru Aleman ^{aleman@maths.lth.se" class="auth_mail" title="E-mail the corresponding author} ; Yurii Lyubarskii ; ^{yura@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author} ; Eugenia Malinnikova ^{eugenia@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author} ; Karl-Mikael Perfekt ^{perfekt@maths.lth.se" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 47B33 ; secondary ; 30H10 ; 30J05 ; 47A45
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 February 2016
• 年：2016
• 卷：270
• 期：3
• 页码：861-883
• 全文大小：442 K

文摘

Let K_ϑ$K_{\vartheta }$ be a model space generated by an inner function ϑ  . We study the Schatten class membership of composition operators C_φ:K_ϑ→H²(D)$C_{\phi }:K_{\vartheta }\to H^2(\mathbb{D})$ with a holomorphic function φ:D→D$\phi :\mathbb{D}\to \mathbb{D}$, and, more generally, of embeddings I_μ:K_θ→L²(μ)$I_{\mu }:K_{\theta }\to L^2(\mu )$ with a positive measure μ   in $\overline{\mathbb{D}}$. In the case of one-component inner functions ϑ we show that the problem can be reduced to the study of natural extensions of I   and C_φ$C_{\phi }$ to the Hardy–Smirnov space E²(D)$E^2(D)$ in some domain D⊃D$D\supset \mathbb{D}$. In particular, we obtain a characterization of Schatten membership of C_φ$C_{\phi }$ in terms of Nevanlinna counting function. By example this characterization does not hold true for general ϑ.