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Extensions of Grothendieck and Bennett–Carl theorems
Grothendieck has proved that the inclusion operator J : l 1 ↪ l 2 is 1-summing. Bennett and Carl proved, independently, that if 1 ≤ p ≤ q ≤ 2 and 1 s = 1 p - 1 q + 1 2 , then the inclusion operator J : l p ↪ l q is ( s , 1) -summing. In this paper we prove the following extension of Grothendieck the...
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| Published in: | Revista matemática complutense 2016-09, Vol.29 (3), p.677-690 |
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| creator | Popa, Dumitru |
| description | Grothendieck has proved that the inclusion operator
J
:
l
1
↪
l
2
is 1-summing. Bennett and Carl proved, independently, that if
1
≤
p
≤
q
≤
2
and
1
s
=
1
p
-
1
q
+
1
2
, then the inclusion operator
J
:
l
p
↪
l
q
is (
s
, 1) -summing. In this paper we prove the following extension of Grothendieck theorem: If
X
,
Y
are Banach spaces and
V
:
X
→
Y
is 1-summing then, the multiplication operator
M
V
:
l
1
[
X
]
→
l
2
(
Y
)
is 1 -summing. Using this result we prove the following extension of Bennett and Carl theorem: If
X
and
Y
are Banach spaces and
V
:
X
→
Y
is 1-summing then, the multiplication operator
M
V
:
l
p
(
X
)
→
l
q
(
Y
)
is (
s
, 1)-summing.
l
1
[
X
]
,
l
p
(
X
)
denotes the Banach spaces of all unconditionally norm convergent series respectively
p
-absolutely convergent series and
M
V
(
(
x
n
)
n
∈
N
)
:
=
(
V
(
x
n
)
)
n
∈
N
is the multiplication operator. |
| doi_str_mv | 10.1007/s13163-016-0198-x |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_1880882819</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1880882819</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-3775a745c3bf10d6fa868f4bbc35328cf7c3099fd2c39aff89351199933dacdb3</originalsourceid><addsrcrecordid>eNp1ULtOAzEQtBBIhMAH0J1EbbBv77Eu4RQCUiQaqC2fz4YLiR3sixQ6_oE_zJfg6ChoKHZnpZ2ZXQ0hl5xdc8bqm8iBV0AZr1IJpLsjMkmINEdWH6eZg6Cp4Sk5i3HJWCkKLCakmu0G42LvXcy8zebBD2_Gdb3R75lyXXZnnDPDsP_6blRYZWnpg1nHc3Ji1Sqai1-ckpf72XPzQBdP88fmdkF1emegUNelqotSQ2s56yqrsEJbtK2GEnLUttbAhLBdrkEoa1FAybkQAqBTumthSq5G303wH1sTB7n02-DSSckRGWKOXCQWH1k6-BiDsXIT-rUKn5IzeYhHjvHIFI88xCN3SZOPmpi47tWEP87_in4AaNdoxw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1880882819</pqid></control><display><type>article</type><title>Extensions of Grothendieck and Bennett–Carl theorems</title><source>Springer Nature</source><creator>Popa, Dumitru</creator><creatorcontrib>Popa, Dumitru</creatorcontrib><description>Grothendieck has proved that the inclusion operator
J
:
l
1
↪
l
2
is 1-summing. Bennett and Carl proved, independently, that if
1
≤
p
≤
q
≤
2
and
1
s
=
1
p
-
1
q
+
1
2
, then the inclusion operator
J
:
l
p
↪
l
q
is (
s
, 1) -summing. In this paper we prove the following extension of Grothendieck theorem: If
X
,
Y
are Banach spaces and
V
:
X
→
Y
is 1-summing then, the multiplication operator
M
V
:
l
1
[
X
]
→
l
2
(
Y
)
is 1 -summing. Using this result we prove the following extension of Bennett and Carl theorem: If
X
and
Y
are Banach spaces and
V
:
X
→
Y
is 1-summing then, the multiplication operator
M
V
:
l
p
(
X
)
→
l
q
(
Y
)
is (
s
, 1)-summing.
l
1
[
X
]
,
l
p
(
X
)
denotes the Banach spaces of all unconditionally norm convergent series respectively
p
-absolutely convergent series and
M
V
(
(
x
n
)
n
∈
N
)
:
=
(
V
(
x
n
)
)
n
∈
N
is the multiplication operator.</description><identifier>ISSN: 1139-1138</identifier><identifier>EISSN: 1988-2807</identifier><identifier>DOI: 10.1007/s13163-016-0198-x</identifier><language>eng</language><publisher>Milan: Springer Milan</publisher><subject>Algebra ; Analysis ; Applications of Mathematics ; Banach spaces ; Convergence ; Error analysis ; Geometry ; Mathematics ; Mathematics and Statistics ; Multiplication ; Multiplication & division ; Theorems ; Topology</subject><ispartof>Revista matemática complutense, 2016-09, Vol.29 (3), p.677-690</ispartof><rights>Universidad Complutense de Madrid 2016</rights><rights>Copyright Springer Science & Business Media 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-3775a745c3bf10d6fa868f4bbc35328cf7c3099fd2c39aff89351199933dacdb3</citedby><cites>FETCH-LOGICAL-c316t-3775a745c3bf10d6fa868f4bbc35328cf7c3099fd2c39aff89351199933dacdb3</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>Popa, Dumitru</creatorcontrib><title>Extensions of Grothendieck and Bennett–Carl theorems</title><title>Revista matemática complutense</title><addtitle>Rev Mat Complut</addtitle><description>Grothendieck has proved that the inclusion operator
J
:
l
1
↪
l
2
is 1-summing. Bennett and Carl proved, independently, that if
1
≤
p
≤
q
≤
2
and
1
s
=
1
p
-
1
q
+
1
2
, then the inclusion operator
J
:
l
p
↪
l
q
is (
s
, 1) -summing. In this paper we prove the following extension of Grothendieck theorem: If
X
,
Y
are Banach spaces and
V
:
X
→
Y
is 1-summing then, the multiplication operator
M
V
:
l
1
[
X
]
→
l
2
(
Y
)
is 1 -summing. Using this result we prove the following extension of Bennett and Carl theorem: If
X
and
Y
are Banach spaces and
V
:
X
→
Y
is 1-summing then, the multiplication operator
M
V
:
l
p
(
X
)
→
l
q
(
Y
)
is (
s
, 1)-summing.
l
1
[
X
]
,
l
p
(
X
)
denotes the Banach spaces of all unconditionally norm convergent series respectively
p
-absolutely convergent series and
M
V
(
(
x
n
)
n
∈
N
)
:
=
(
V
(
x
n
)
)
n
∈
N
is the multiplication operator.</description><subject>Algebra</subject><subject>Analysis</subject><subject>Applications of Mathematics</subject><subject>Banach spaces</subject><subject>Convergence</subject><subject>Error analysis</subject><subject>Geometry</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Multiplication</subject><subject>Multiplication & division</subject><subject>Theorems</subject><subject>Topology</subject><issn>1139-1138</issn><issn>1988-2807</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1ULtOAzEQtBBIhMAH0J1EbbBv77Eu4RQCUiQaqC2fz4YLiR3sixQ6_oE_zJfg6ChoKHZnpZ2ZXQ0hl5xdc8bqm8iBV0AZr1IJpLsjMkmINEdWH6eZg6Cp4Sk5i3HJWCkKLCakmu0G42LvXcy8zebBD2_Gdb3R75lyXXZnnDPDsP_6blRYZWnpg1nHc3Ji1Sqai1-ckpf72XPzQBdP88fmdkF1emegUNelqotSQ2s56yqrsEJbtK2GEnLUttbAhLBdrkEoa1FAybkQAqBTumthSq5G303wH1sTB7n02-DSSckRGWKOXCQWH1k6-BiDsXIT-rUKn5IzeYhHjvHIFI88xCN3SZOPmpi47tWEP87_in4AaNdoxw</recordid><startdate>20160901</startdate><enddate>20160901</enddate><creator>Popa, Dumitru</creator><general>Springer Milan</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20160901</creationdate><title>Extensions of Grothendieck and Bennett–Carl theorems</title><author>Popa, Dumitru</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-3775a745c3bf10d6fa868f4bbc35328cf7c3099fd2c39aff89351199933dacdb3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Algebra</topic><topic>Analysis</topic><topic>Applications of Mathematics</topic><topic>Banach spaces</topic><topic>Convergence</topic><topic>Error analysis</topic><topic>Geometry</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Multiplication</topic><topic>Multiplication & division</topic><topic>Theorems</topic><topic>Topology</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Popa, Dumitru</creatorcontrib><collection>CrossRef</collection><jtitle>Revista matemática complutense</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Popa, Dumitru</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Extensions of Grothendieck and Bennett–Carl theorems</atitle><jtitle>Revista matemática complutense</jtitle><stitle>Rev Mat Complut</stitle><date>2016-09-01</date><risdate>2016</risdate><volume>29</volume><issue>3</issue><spage>677</spage><epage>690</epage><pages>677-690</pages><issn>1139-1138</issn><eissn>1988-2807</eissn><abstract>Grothendieck has proved that the inclusion operator
J
:
l
1
↪
l
2
is 1-summing. Bennett and Carl proved, independently, that if
1
≤
p
≤
q
≤
2
and
1
s
=
1
p
-
1
q
+
1
2
, then the inclusion operator
J
:
l
p
↪
l
q
is (
s
, 1) -summing. In this paper we prove the following extension of Grothendieck theorem: If
X
,
Y
are Banach spaces and
V
:
X
→
Y
is 1-summing then, the multiplication operator
M
V
:
l
1
[
X
]
→
l
2
(
Y
)
is 1 -summing. Using this result we prove the following extension of Bennett and Carl theorem: If
X
and
Y
are Banach spaces and
V
:
X
→
Y
is 1-summing then, the multiplication operator
M
V
:
l
p
(
X
)
→
l
q
(
Y
)
is (
s
, 1)-summing.
l
1
[
X
]
,
l
p
(
X
)
denotes the Banach spaces of all unconditionally norm convergent series respectively
p
-absolutely convergent series and
M
V
(
(
x
n
)
n
∈
N
)
:
=
(
V
(
x
n
)
)
n
∈
N
is the multiplication operator.</abstract><cop>Milan</cop><pub>Springer Milan</pub><doi>10.1007/s13163-016-0198-x</doi><tpages>14</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1139-1138 |
| ispartof | Revista matemática complutense, 2016-09, Vol.29 (3), p.677-690 |
| issn | 1139-1138 1988-2807 |
| language | eng |
| recordid | cdi_proquest_journals_1880882819 |
| source | Springer Nature |
| subjects | Algebra Analysis Applications of Mathematics Banach spaces Convergence Error analysis Geometry Mathematics Mathematics and Statistics Multiplication Multiplication & division Theorems Topology |
| title | Extensions of Grothendieck and Bennett–Carl theorems |
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