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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si1.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=dfad4a7fb843a20103aa10b5be5e0e1e" title="Click to view the MathML source">K_ϑclass="mathContainer hidden">class="mathCode">$K_{\vartheta }$ be a model space generated by an inner function ϑ  . We study the Schatten class membership of composition operators class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si2.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=63548c6968764d51449dce89ded5e18a" title="Click to view the MathML source">C_φ:K_ϑ→H²(D)class="mathContainer hidden">class="mathCode">$C_{\phi }:K_{\vartheta }\to H^2(\mathbb{D})$ with a holomorphic function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si20.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=d6efdfd90d4bc23731c5bd94b05f40ff" title="Click to view the MathML source">φ:D→Dclass="mathContainer hidden">class="mathCode">$\phi :\mathbb{D}\to \mathbb{D}$, and, more generally, of embeddings class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si4.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=7d6c1a10cbd314dff7843a5a57d0c5b4" title="Click to view the MathML source">I_μ:K_θ→L²(μ)class="mathContainer hidden">class="mathCode">$I_{\mu }:K_{\theta }\to L^2(\mu )$ with a positive measure μ   in class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si5.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=b896bfc72807ecc3895edffa3fcc155b">class="imgLazyJSB inlineImage" height="15" width="14" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123615004577-si5.gif">class="mathContainer hidden">class="mathCode">$\overline{\mathbb{D}}$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si6.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=6dc8dc4c25f403f616e2d4370814a791" title="Click to view the MathML source">C_φclass="mathContainer hidden">class="mathCode">$C_{\phi }$ to the Hardy–Smirnov space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si7.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=9f126bf538484c003b96ff9dff7f980b" title="Click to view the MathML source">E²(D)class="mathContainer hidden">class="mathCode">$E^2(D)$ in some domain class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si8.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=d5248f8f42091317f6b25a06b5deba14" title="Click to view the MathML source">D⊃Dclass="mathContainer hidden">class="mathCode">$D\supset \mathbb{D}$. In particular, we obtain a characterization of Schatten membership of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123615004577&_mathId=si6.gif&_user=111111111&_pii=S0022123615004577&_rdoc=1&_issn=00221236&md5=6dc8dc4c25f403f616e2d4370814a791" title="Click to view the MathML source">C_φclass="mathContainer hidden">class="mathCode">$C_{\phi }$ in terms of Nevanlinna counting function. By example this characterization does not hold true for general ϑ.