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 id="mmlsi1" class="mathmlsrc">ixSupport 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^⁎iner hidden">i1.gif" overflow="scroll">i>Ci>⁎-algebras generated by Toeplitz operators acting on weighted Bergman spaces id="mmlsi106" class="mathmlsrc">itle="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"><img 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">ipt><img height="19" border="0" style="vertical-align:bottom" width="55" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022123616302695-si106.gif">ipt>iner hidden">i106.gif" overflow="scroll">i mathvariant="bold-script">Ai>i>λi>2(i mathvariant="double-struck">Bi>2) over the complex unit ball id="mmlsi3" class="mathmlsrc">ixSupport 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⊂C²iner hidden">i3.gif" overflow="scroll">i mathvariant="double-struck">Bi>2⊂i mathvariant="double-struck">Ci>2. Our key point is an orthogonal decomposition of id="mmlsi106" class="mathmlsrc">itle="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"><img 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">ipt><img height="19" border="0" style="vertical-align:bottom" width="55" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022123616302695-si106.gif">ipt>iner hidden">i106.gif" overflow="scroll">i mathvariant="bold-script">Ai>i>λi>2(i mathvariant="double-struck">Bi>2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space id="mmlsi4" class="mathmlsrc">itle="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"><img 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">ipt><img height="21" border="0" style="vertical-align:bottom" width="47" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022123616302695-si4.gif">ipt>iner hidden">i4.gif" overflow="scroll">i mathvariant="script">Ai>i>μi>2(i mathvariant="double-struck">Di>) over the complex unit disk id="mmlsi5" class="mathmlsrc">ixSupport 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">Diner hidden">i5.gif" overflow="scroll">i mathvariant="double-struck">Di>. 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 id="mmlsi4" class="mathmlsrc">itle="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"><img 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">ipt><img height="21" border="0" style="vertical-align:bottom" width="47" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022123616302695-si4.gif">ipt>iner hidden">i4.gif" overflow="scroll">i mathvariant="script">Ai>i>μi>2(i mathvariant="double-struck">Di>).

id="sp0020">The symbols of the generating Toeplitz operators are chosen to be suitable extensions to id="mmlsi6" class="mathmlsrc">ixSupport 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">B²iner hidden">i6.gif" overflow="scroll">i mathvariant="double-struck">Bi>2 of families id="mmlsi7" class="mathmlsrc">ixSupport 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">Siner hidden">i7.gif" overflow="scroll">i mathvariant="script">Si> of bounded functions on id="mmlsi5" class="mathmlsrc">ixSupport 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">Diner hidden">i5.gif" overflow="scroll">i mathvariant="double-struck">Di>. Symbol classes id="mmlsi7" class="mathmlsrc">ixSupport 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">Siner hidden">i7.gif" overflow="scroll">i mathvariant="script">Si> that generate important classical commutative and non-commutative Toeplitz algebras in id="mmlsi38" class="mathmlsrc">itle="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"><img 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">ipt><img height="21" border="0" style="vertical-align:bottom" width="70" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022123616302695-si38.gif">ipt>iner hidden">i38.gif" overflow="scroll">i mathvariant="script">Li>(i mathvariant="script">Ai>i>μi>2(i mathvariant="double-struck">Di>)) are of particular interest. In this paper we discuss various examples. In the case of id="mmlsi9" class="mathmlsrc">itle="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"><img 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">ipt><img height="18" border="0" style="vertical-align:bottom" width="71" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022123616302695-si9.gif">ipt>iner hidden">i9.gif" overflow="scroll">i mathvariant="script">Si>=i>Ci>(i mathvariant="double-struck">Di>&oline;) and id="mmlsi10" class="mathmlsrc">itle="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"><img 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">ipt><img height="18" border="0" style="vertical-align:bottom" width="151" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022123616302695-si10.gif">ipt>iner hidden">i10.gif" overflow="scroll">i mathvariant="script">Si>=i>Ci>(i mathvariant="double-struck">Di>&oline;)&otimes;i>Li>&infin;(0,1) we characterize all irreducible representations of the resulting Toeplitz operator id="mmlsi1" class="mathmlsrc">ixSupport 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^⁎iner hidden">i1.gif" overflow="scroll">i>Ci>⁎-algebras. Their Calkin algebras are described and index formulas are provided.