On algebras generated by Toeplitz operators and their representations

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• 作者：Wolfram Bauer^a ; ^{bauer@math.uni-hannover.de" class="auth_mail" title="E-mail the corresponding author} ; Nikolai Vasilevski^b ; ^{nvasilev@math.cinvestav.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 47B35 ; 47L80 ; secondary ; 32A36
• 刊名：Journal of Functional Analysis
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：272
• 期：2
• 页码：705-737
• 全文大小：568 K

文摘

We study Banach and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si1.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=bc2d4370a8d4d35557d1cba9a54ff005" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebras generated by Toeplitz operators acting on weighted Bergman spaces class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si106.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=4d0ca79b1680ad292875aea786de7fc7">class="imgLazyJSB inlineImage" height="19" width="55" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si106.gif">class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\lambda }^2({\mathbb{B}}^2)$ over the complex unit ball class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si3.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=50a3a3cfbe4bf2f391edea20ae2674a6" title="Click to view the MathML source">B²⊂C²class="mathContainer hidden">class="mathCode">${\mathbb{B}}^2\subset {\mathbb{C}}^2$. Our key point is an orthogonal decomposition of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si106.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=4d0ca79b1680ad292875aea786de7fc7">class="imgLazyJSB inlineImage" height="19" width="55" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si106.gif">class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\lambda }^2({\mathbb{B}}^2)$ into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si4.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=1b6f6fa46a59f90a18aff8ab4950ff4b">class="imgLazyJSB inlineImage" height="21" width="47" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si4.gif">class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\mu }^2(\mathbb{D})$ over the complex unit disk class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si5.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=8194308aa1939c159e2717739dde9fbd" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$\mathbb{D}$. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si4.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=1b6f6fa46a59f90a18aff8ab4950ff4b">class="imgLazyJSB inlineImage" height="21" width="47" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si4.gif">class="mathContainer hidden">class="mathCode">${\mathcal{A}}_{\mu }^2(\mathbb{D})$.

The symbols of the generating Toeplitz operators are chosen to be suitable extensions to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si6.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=4143882a8ed90c134c8d437fec2b5699" title="Click to view the MathML source">B²class="mathContainer hidden">class="mathCode">${\mathbb{B}}^2$ of families class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si7.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=7de67b4ab5872ac9c465fdf222e9c10b" title="Click to view the MathML source">Sclass="mathContainer hidden">class="mathCode">$\mathcal{S}$ of bounded functions on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si5.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=8194308aa1939c159e2717739dde9fbd" title="Click to view the MathML source">Dclass="mathContainer hidden">class="mathCode">$\mathbb{D}$. Symbol classes class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si7.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=7de67b4ab5872ac9c465fdf222e9c10b" title="Click to view the MathML source">Sclass="mathContainer hidden">class="mathCode">$\mathcal{S}$ that generate important classical commutative and non-commutative Toeplitz algebras in class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si38.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=f75ca6f234696a0b92870e2dc47bd1f5">class="imgLazyJSB inlineImage" height="21" width="70" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si38.gif">class="mathContainer hidden">class="mathCode">$\mathcal{L}({\mathcal{A}}_{\mu }^2(\mathbb{D}))$ are of particular interest. In this paper we discuss various examples. In the case of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si9.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=e5f13752e05d1a391135e630db690b9a">class="imgLazyJSB inlineImage" height="18" width="71" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si9.gif">class="mathContainer hidden">class="mathCode">$\mathcal{S}=C(\bar{\mathbb{D}})$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si10.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=746bc3f6164d8c5d53097bf83fed887a">class="imgLazyJSB inlineImage" height="18" width="151" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302695-si10.gif">class="mathContainer hidden">class="mathCode">$\mathcal{S}=C(\bar{\mathbb{D}})\otimes L_{\infty }(0,1)$ we characterize all irreducible representations of the resulting Toeplitz operator class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302695&_mathId=si1.gif&_user=111111111&_pii=S0022123616302695&_rdoc=1&_issn=00221236&md5=bc2d4370a8d4d35557d1cba9a54ff005" title="Click to view the MathML source">C^⁎class="mathContainer hidden">class="mathCode">$C^{⁎}$-algebras. Their Calkin algebras are described and index formulas are provided.