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 mmlsi1" class="mathmlsrc">mulatext 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^⁎mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msup><mrow><mi>Cmi>mrow><mrow><mo>⁎mo>mrow>msup>math>-algebras generated by Toeplitz operators acting on weighted Bergman spaces mmlsi106" class="mathmlsrc">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">mg 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">mathContainer hidden">mathCode"><math altimg="si106.gif" overflow="scroll"><msubsup><mrow><mi mathvariant="bold-script">Ami>mrow><mrow><mi>λmi>mrow><mrow><mn>2mn>mrow>msubsup><mo stretchy="false">(mo><msup><mrow><mi mathvariant="double-struck">Bmi>mrow><mrow><mn>2mn>mrow>msup><mo stretchy="false">)mo>math> over the complex unit ball mmlsi3" class="mathmlsrc">mulatext 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⊂C²mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><msup><mrow><mi mathvariant="double-struck">Bmi>mrow><mrow><mn>2mn>mrow>msup><mo>⊂mo><msup><mrow><mi mathvariant="double-struck">Cmi>mrow><mrow><mn>2mn>mrow>msup>math>. Our key point is an orthogonal decomposition of mmlsi106" class="mathmlsrc">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">mg 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">mathContainer hidden">mathCode"><math altimg="si106.gif" overflow="scroll"><msubsup><mrow><mi mathvariant="bold-script">Ami>mrow><mrow><mi>λmi>mrow><mrow><mn>2mn>mrow>msubsup><mo stretchy="false">(mo><msup><mrow><mi mathvariant="double-struck">Bmi>mrow><mrow><mn>2mn>mrow>msup><mo stretchy="false">)mo>math> into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space mmlsi4" class="mathmlsrc">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">mg 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">mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msubsup><mrow><mi mathvariant="script">Ami>mrow><mrow><mi>μmi>mrow><mrow><mn>2mn>mrow>msubsup><mo stretchy="false">(mo><mi mathvariant="double-struck">Dmi><mo stretchy="false">)mo>math> over the complex unit disk mmlsi5" class="mathmlsrc">mulatext 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">DmathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><mi mathvariant="double-struck">Dmi>math>. 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 mmlsi4" class="mathmlsrc">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">mg 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">mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msubsup><mrow><mi mathvariant="script">Ami>mrow><mrow><mi>μmi>mrow><mrow><mn>2mn>mrow>msubsup><mo stretchy="false">(mo><mi mathvariant="double-struck">Dmi><mo stretchy="false">)mo>math>.

The symbols of the generating Toeplitz operators are chosen to be suitable extensions to mmlsi6" class="mathmlsrc">mulatext 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">B²mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><msup><mrow><mi mathvariant="double-struck">Bmi>mrow><mrow><mn>2mn>mrow>msup>math> of families mmlsi7" class="mathmlsrc">mulatext 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">SmathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mi mathvariant="script">Smi>math> of bounded functions on mmlsi5" class="mathmlsrc">mulatext 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">DmathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><mi mathvariant="double-struck">Dmi>math>. Symbol classes mmlsi7" class="mathmlsrc">mulatext 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">SmathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><mi mathvariant="script">Smi>math> that generate important classical commutative and non-commutative Toeplitz algebras in mmlsi38" class="mathmlsrc">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">mg 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">mathContainer hidden">mathCode"><math altimg="si38.gif" overflow="scroll"><mi mathvariant="script">Lmi><mo stretchy="false">(mo><msubsup><mrow><mi mathvariant="script">Ami>mrow><mrow><mi>μmi>mrow><mrow><mn>2mn>mrow>msubsup><mo stretchy="false">(mo><mi mathvariant="double-struck">Dmi><mo stretchy="false">)mo><mo stretchy="false">)mo>math> are of particular interest. In this paper we discuss various examples. In the case of mmlsi9" class="mathmlsrc">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">mg 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">mathContainer hidden">mathCode"><math altimg="si9.gif" overflow="scroll"><mi mathvariant="script">Smi><mo>=mo><mi>Cmi><mo stretchy="false">(mo><mover accent="true"><mrow><mi mathvariant="double-struck">Dmi>mrow><mo>‾mo>mover><mo stretchy="false">)mo>math> and mmlsi10" class="mathmlsrc">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">mg 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">mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll"><mi mathvariant="script">Smi><mo>=mo><mi>Cmi><mo stretchy="false">(mo><mover accent="true"><mrow><mi mathvariant="double-struck">Dmi>mrow><mo>‾mo>mover><mo stretchy="false">)mo><mo>&otimes;mo><msub><mrow><mi>Lmi>mrow><mrow><mo>∞mo>mrow>msub><mo stretchy="false">(mo><mn>0mn><mo>,mo><mn>1mn><mo stretchy="false">)mo>math> we characterize all irreducible representations of the resulting Toeplitz operator mmlsi1" class="mathmlsrc">mulatext 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^⁎mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msup><mrow><mi>Cmi>mrow><mrow><mo>⁎mo>mrow>msup>math>-algebras. Their Calkin algebras are described and index formulas are provided.