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Grothendieck‐Lidskiǐ theorem for subspaces of L p ‐spaces
In 1955, A. Grothendieck has shown that if the linear operator T in a Banach subspace of an L ∞ ‐space is 2/3‐nuclear then the trace of T is well defined and is equal to the sum of all eigenvalues {μ k ( T )} of T . Lidskiǐ, in 1959, proved his famous theorem on the coincidence of the trace of the S...
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| Published in: | Mathematische Nachrichten 2013-02, Vol.286 (2-3), p.279-282 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| container_end_page | 282 |
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| container_title | Mathematische Nachrichten |
| container_volume | 286 |
| creator | Reinov, Oleg Latif, Qaisar |
| description | In 1955, A. Grothendieck has shown that if the linear operator
T
in a Banach subspace of an
L
∞
‐space is 2/3‐nuclear then the trace of
T
is well defined and is equal to the sum of all eigenvalues {μ
k
(
T
)} of
T
. Lidskiǐ, in 1959, proved his famous theorem on the coincidence of the trace of the
S
1
‐operator in
L
2
(ν) with its spectral trace
\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sum _{k=1}^\infty \mu _k(T)$\end{document}
. We show that
for p ∈ [1, ∞] and s ∈ (0, 1] with 1/s = 1 + |1/2 − 1/p|, and for every s‐nuclear operator T in every subspace of any L
p
(ν)‐space the trace of T is well defined and equals the sum of all eigenvalues of T.
Note that for
p
= 2 one has
s
= 1, and for
p
= ∞ one has
s
= 2/3. |
| doi_str_mv | 10.1002/mana.201100112 |
| format | article |
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T
in a Banach subspace of an
L
∞
‐space is 2/3‐nuclear then the trace of
T
is well defined and is equal to the sum of all eigenvalues {μ
k
(
T
)} of
T
. Lidskiǐ, in 1959, proved his famous theorem on the coincidence of the trace of the
S
1
‐operator in
L
2
(ν) with its spectral trace
\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sum _{k=1}^\infty \mu _k(T)$\end{document}
. We show that
for p ∈ [1, ∞] and s ∈ (0, 1] with 1/s = 1 + |1/2 − 1/p|, and for every s‐nuclear operator T in every subspace of any L
p
(ν)‐space the trace of T is well defined and equals the sum of all eigenvalues of T.
Note that for
p
= 2 one has
s
= 1, and for
p
= ∞ one has
s
= 2/3.</description><identifier>ISSN: 0025-584X</identifier><identifier>EISSN: 1522-2616</identifier><identifier>DOI: 10.1002/mana.201100112</identifier><language>eng</language><ispartof>Mathematische Nachrichten, 2013-02, Vol.286 (2-3), p.279-282</ispartof><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c169t-6a5678a967c1d611e2510643ff98dd0fddc66d8df8eaa802d94357f45d2514553</citedby><cites>FETCH-LOGICAL-c169t-6a5678a967c1d611e2510643ff98dd0fddc66d8df8eaa802d94357f45d2514553</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids></links><search><creatorcontrib>Reinov, Oleg</creatorcontrib><creatorcontrib>Latif, Qaisar</creatorcontrib><title>Grothendieck‐Lidskiǐ theorem for subspaces of L p ‐spaces</title><title>Mathematische Nachrichten</title><description>In 1955, A. Grothendieck has shown that if the linear operator
T
in a Banach subspace of an
L
∞
‐space is 2/3‐nuclear then the trace of
T
is well defined and is equal to the sum of all eigenvalues {μ
k
(
T
)} of
T
. Lidskiǐ, in 1959, proved his famous theorem on the coincidence of the trace of the
S
1
‐operator in
L
2
(ν) with its spectral trace
\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sum _{k=1}^\infty \mu _k(T)$\end{document}
. We show that
for p ∈ [1, ∞] and s ∈ (0, 1] with 1/s = 1 + |1/2 − 1/p|, and for every s‐nuclear operator T in every subspace of any L
p
(ν)‐space the trace of T is well defined and equals the sum of all eigenvalues of T.
Note that for
p
= 2 one has
s
= 1, and for
p
= ∞ one has
s
= 2/3.</description><issn>0025-584X</issn><issn>1522-2616</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNo9j8FKxDAURYMoWEe3rvMDre-lTZpsBBl0Rii4mQF3JeYlWMdOSzIu3PkJ81f-h19ihxFXl3s5XDiMXSMUCCBueru1hQCcCqI4YRlKIXKhUJ2ybAJkLnX1fM4uUnoDAGNqlbHbRRx2r35LnXebn69901HadN97Po1D9D0PQ-Tp4yWN1vnEh8AbPvIJPA6X7CzY9-Sv_nLG1g_3q_kyb54Wj_O7JneozC5XVqpaW6Nqh6QQvZAIqipDMJoIApFTijQF7a3VIMhUpaxDJWkCKynLGSuOvy4OKUUf2jF2vY2fLUJ7sG8P9u2_ffkLR19PkQ</recordid><startdate>201302</startdate><enddate>201302</enddate><creator>Reinov, Oleg</creator><creator>Latif, Qaisar</creator><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>201302</creationdate><title>Grothendieck‐Lidskiǐ theorem for subspaces of L p ‐spaces</title><author>Reinov, Oleg ; Latif, Qaisar</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c169t-6a5678a967c1d611e2510643ff98dd0fddc66d8df8eaa802d94357f45d2514553</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Reinov, Oleg</creatorcontrib><creatorcontrib>Latif, Qaisar</creatorcontrib><collection>CrossRef</collection><jtitle>Mathematische Nachrichten</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Reinov, Oleg</au><au>Latif, Qaisar</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Grothendieck‐Lidskiǐ theorem for subspaces of L p ‐spaces</atitle><jtitle>Mathematische Nachrichten</jtitle><date>2013-02</date><risdate>2013</risdate><volume>286</volume><issue>2-3</issue><spage>279</spage><epage>282</epage><pages>279-282</pages><issn>0025-584X</issn><eissn>1522-2616</eissn><abstract>In 1955, A. Grothendieck has shown that if the linear operator
T
in a Banach subspace of an
L
∞
‐space is 2/3‐nuclear then the trace of
T
is well defined and is equal to the sum of all eigenvalues {μ
k
(
T
)} of
T
. Lidskiǐ, in 1959, proved his famous theorem on the coincidence of the trace of the
S
1
‐operator in
L
2
(ν) with its spectral trace
\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sum _{k=1}^\infty \mu _k(T)$\end{document}
. We show that
for p ∈ [1, ∞] and s ∈ (0, 1] with 1/s = 1 + |1/2 − 1/p|, and for every s‐nuclear operator T in every subspace of any L
p
(ν)‐space the trace of T is well defined and equals the sum of all eigenvalues of T.
Note that for
p
= 2 one has
s
= 1, and for
p
= ∞ one has
s
= 2/3.</abstract><doi>10.1002/mana.201100112</doi><tpages>4</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0025-584X |
| ispartof | Mathematische Nachrichten, 2013-02, Vol.286 (2-3), p.279-282 |
| issn | 0025-584X 1522-2616 |
| language | eng |
| recordid | cdi_crossref_primary_10_1002_mana_201100112 |
| source | Wiley |
| title | Grothendieck‐Lidskiǐ theorem for subspaces of L p ‐spaces |
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