The Essential Spectrum of Toeplitz Operators on the Unit Ball
In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces A ν p ( B n ) , where p ∈ ( 1 , ∞ ) and B n ⊂ C n denotes the n -dimensional open unit ball. Let f be a continuous function on the Euclidean closure of B n . It is well-known that then the correspo...
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description | In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces
A
ν
p
(
B
n
)
, where
p
∈
(
1
,
∞
)
and
B
n
⊂
C
n
denotes the
n
-dimensional open unit ball. Let
f
be a continuous function on the Euclidean closure of
B
n
. It is well-known that then the corresponding Toeplitz operator
T
f
is Fredholm if and only if
f
has no zeros on the boundary
∂
B
n
. As a consequence, the essential spectrum of
T
f
is given by the boundary values of
f
. We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suárez et al. (Integral Equ Oper Theory 75:197–233,
2013
, Indiana Univ Math J 56(5):2185–2232,
2007
) and limit operator techniques coming from similar problems on the sequence space
ℓ
p
(
Z
)
(Hagger et al. in J Math Anal Appl 437(1):255–291,
2016
; Lindner and Seidel in J Funct Anal 267(3):901–917,
2014
; Rabinovich et al. Integral Equ Oper Theory 30(4): 452–495,
1998
and references therein). |
doi_str_mv | 10.1007/s00020-017-2399-1 |
format | Article |
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A
ν
p
(
B
n
)
, where
p
∈
(
1
,
∞
)
and
B
n
⊂
C
n
denotes the
n
-dimensional open unit ball. Let
f
be a continuous function on the Euclidean closure of
B
n
. It is well-known that then the corresponding Toeplitz operator
T
f
is Fredholm if and only if
f
has no zeros on the boundary
∂
B
n
. As a consequence, the essential spectrum of
T
f
is given by the boundary values of
f
. We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suárez et al. (Integral Equ Oper Theory 75:197–233,
2013
, Indiana Univ Math J 56(5):2185–2232,
2007
) and limit operator techniques coming from similar problems on the sequence space
ℓ
p
(
Z
)
(Hagger et al. in J Math Anal Appl 437(1):255–291,
2016
; Lindner and Seidel in J Funct Anal 267(3):901–917,
2014
; Rabinovich et al. Integral Equ Oper Theory 30(4): 452–495,
1998
and references therein).</description><identifier>ISSN: 0378-620X</identifier><identifier>EISSN: 1420-8989</identifier><identifier>DOI: 10.1007/s00020-017-2399-1</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Analysis ; Continuity (mathematics) ; Economic impact ; Integrals ; Mathematics ; Mathematics and Statistics ; Operators (mathematics)</subject><ispartof>Integral equations and operator theory, 2017-12, Vol.89 (4), p.519-556</ispartof><rights>Springer International Publishing AG 2017</rights><rights>Copyright Springer Science & Business Media 2017</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-ac73dfc836e8cbc017b9fbe33ebccb4732af882754822070f12baea1832a33a93</citedby><cites>FETCH-LOGICAL-c316t-ac73dfc836e8cbc017b9fbe33ebccb4732af882754822070f12baea1832a33a93</cites><orcidid>0000-0002-4289-3025</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00020-017-2399-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00020-017-2399-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>Hagger, Raffael</creatorcontrib><title>The Essential Spectrum of Toeplitz Operators on the Unit Ball</title><title>Integral equations and operator theory</title><addtitle>Integr. Equ. Oper. Theory</addtitle><description>In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces
A
ν
p
(
B
n
)
, where
p
∈
(
1
,
∞
)
and
B
n
⊂
C
n
denotes the
n
-dimensional open unit ball. Let
f
be a continuous function on the Euclidean closure of
B
n
. It is well-known that then the corresponding Toeplitz operator
T
f
is Fredholm if and only if
f
has no zeros on the boundary
∂
B
n
. As a consequence, the essential spectrum of
T
f
is given by the boundary values of
f
. We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suárez et al. (Integral Equ Oper Theory 75:197–233,
2013
, Indiana Univ Math J 56(5):2185–2232,
2007
) and limit operator techniques coming from similar problems on the sequence space
ℓ
p
(
Z
)
(Hagger et al. in J Math Anal Appl 437(1):255–291,
2016
; Lindner and Seidel in J Funct Anal 267(3):901–917,
2014
; Rabinovich et al. Integral Equ Oper Theory 30(4): 452–495,
1998
and references therein).</description><subject>Analysis</subject><subject>Continuity (mathematics)</subject><subject>Economic impact</subject><subject>Integrals</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operators (mathematics)</subject><issn>0378-620X</issn><issn>1420-8989</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kE9PAyEQxYnRxFr9AN5IPKMDtAscPGhT_yRNenCbeCMsgm6zXVagB_300qwHL54mmfm9Ny8PoUsK1xRA3CQAYECACsK4UoQeoQmdlY1UUh2jCXAhScXg9RSdpbQtMBOsmqDb-sPhZUquz63p8MvgbI77HQ4e18ENXZu_8Xpw0eQQEw49zoXf9G3G96brztGJN11yF79zijYPy3rxRFbrx-fF3YpYTqtMjBX8zVvJKydtY0vIRvnGce4aa5uZ4Mx4KZmYzyRjIMBT1hhnqCwHzo3iU3Q1-g4xfO5dynob9rEvLzVVAjhwxueFoiNlY0gpOq-H2O5M_NIU9KElPbakSwB9aEnTomGjJhW2f3fxj_O_oh8j3Gkq</recordid><startdate>20171201</startdate><enddate>20171201</enddate><creator>Hagger, Raffael</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-4289-3025</orcidid></search><sort><creationdate>20171201</creationdate><title>The Essential Spectrum of Toeplitz Operators on the Unit Ball</title><author>Hagger, Raffael</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-ac73dfc836e8cbc017b9fbe33ebccb4732af882754822070f12baea1832a33a93</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Analysis</topic><topic>Continuity (mathematics)</topic><topic>Economic impact</topic><topic>Integrals</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operators (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Hagger, Raffael</creatorcontrib><collection>CrossRef</collection><jtitle>Integral equations and operator theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Hagger, Raffael</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Essential Spectrum of Toeplitz Operators on the Unit Ball</atitle><jtitle>Integral equations and operator theory</jtitle><stitle>Integr. Equ. Oper. Theory</stitle><date>2017-12-01</date><risdate>2017</risdate><volume>89</volume><issue>4</issue><spage>519</spage><epage>556</epage><pages>519-556</pages><issn>0378-620X</issn><eissn>1420-8989</eissn><abstract>In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces
A
ν
p
(
B
n
)
, where
p
∈
(
1
,
∞
)
and
B
n
⊂
C
n
denotes the
n
-dimensional open unit ball. Let
f
be a continuous function on the Euclidean closure of
B
n
. It is well-known that then the corresponding Toeplitz operator
T
f
is Fredholm if and only if
f
has no zeros on the boundary
∂
B
n
. As a consequence, the essential spectrum of
T
f
is given by the boundary values of
f
. We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suárez et al. (Integral Equ Oper Theory 75:197–233,
2013
, Indiana Univ Math J 56(5):2185–2232,
2007
) and limit operator techniques coming from similar problems on the sequence space
ℓ
p
(
Z
)
(Hagger et al. in J Math Anal Appl 437(1):255–291,
2016
; Lindner and Seidel in J Funct Anal 267(3):901–917,
2014
; Rabinovich et al. Integral Equ Oper Theory 30(4): 452–495,
1998
and references therein).</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00020-017-2399-1</doi><tpages>38</tpages><orcidid>https://orcid.org/0000-0002-4289-3025</orcidid></addata></record> |
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subjects | Analysis Continuity (mathematics) Economic impact Integrals Mathematics Mathematics and Statistics Operators (mathematics) |
title | The Essential Spectrum of Toeplitz Operators on the Unit Ball |
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