Weak precompactness in projective tensor products
We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever...
Ausführliche Beschreibung
Autor*in: |
Rodríguez, José [verfasserIn] Rueda Zoca, Abraham [verfasserIn] |
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Format: |
E-Artikel |
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Sprache: |
Englisch |
Erschienen: |
2023 |
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Schlagwörter: |
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Übergeordnetes Werk: |
Enthalten in: Indagationes mathematicae - Amsterdam : Elsevier, 1990, 35, Seite 60-75 |
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Übergeordnetes Werk: |
volume:35 ; pages:60-75 |
DOI / URN: |
10.1016/j.indag.2023.08.003 |
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Katalog-ID: |
ELV06643288X |
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245 | 1 | 0 | |a Weak precompactness in projective tensor products |
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520 | |a We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . | ||
650 | 4 | |a Projective tensor product | |
650 | 4 | |a Weakly compact set | |
650 | 4 | |a Weakly precompact set | |
650 | 4 | |a Coarse | |
700 | 1 | |a Rueda Zoca, Abraham |e verfasserin |4 aut | |
773 | 0 | 8 | |i Enthalten in |t Indagationes mathematicae |d Amsterdam : Elsevier, 1990 |g 35, Seite 60-75 |h Online-Ressource |w (DE-627)265549957 |w (DE-600)1465401-5 |w (DE-576)11281512X |7 nnns |
773 | 1 | 8 | |g volume:35 |g pages:60-75 |
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936 | b | k | |a 31.00 |j Mathematik: Allgemeines |q VZ |
951 | |a AR | ||
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publishDate |
2023 |
allfields |
10.1016/j.indag.2023.08.003 doi (DE-627)ELV06643288X (ELSEVIER)S0019-3577(23)00080-0 DE-627 ger DE-627 rda eng 510 VZ 31.00 bkl Rodríguez, José verfasserin aut Weak precompactness in projective tensor products 2023 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . Projective tensor product Weakly compact set Weakly precompact set Coarse Rueda Zoca, Abraham verfasserin aut Enthalten in Indagationes mathematicae Amsterdam : Elsevier, 1990 35, Seite 60-75 Online-Ressource (DE-627)265549957 (DE-600)1465401-5 (DE-576)11281512X nnns volume:35 pages:60-75 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.00 Mathematik: Allgemeines VZ AR 35 60-75 |
spelling |
10.1016/j.indag.2023.08.003 doi (DE-627)ELV06643288X (ELSEVIER)S0019-3577(23)00080-0 DE-627 ger DE-627 rda eng 510 VZ 31.00 bkl Rodríguez, José verfasserin aut Weak precompactness in projective tensor products 2023 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . Projective tensor product Weakly compact set Weakly precompact set Coarse Rueda Zoca, Abraham verfasserin aut Enthalten in Indagationes mathematicae Amsterdam : Elsevier, 1990 35, Seite 60-75 Online-Ressource (DE-627)265549957 (DE-600)1465401-5 (DE-576)11281512X nnns volume:35 pages:60-75 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.00 Mathematik: Allgemeines VZ AR 35 60-75 |
allfields_unstemmed |
10.1016/j.indag.2023.08.003 doi (DE-627)ELV06643288X (ELSEVIER)S0019-3577(23)00080-0 DE-627 ger DE-627 rda eng 510 VZ 31.00 bkl Rodríguez, José verfasserin aut Weak precompactness in projective tensor products 2023 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . Projective tensor product Weakly compact set Weakly precompact set Coarse Rueda Zoca, Abraham verfasserin aut Enthalten in Indagationes mathematicae Amsterdam : Elsevier, 1990 35, Seite 60-75 Online-Ressource (DE-627)265549957 (DE-600)1465401-5 (DE-576)11281512X nnns volume:35 pages:60-75 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.00 Mathematik: Allgemeines VZ AR 35 60-75 |
allfieldsGer |
10.1016/j.indag.2023.08.003 doi (DE-627)ELV06643288X (ELSEVIER)S0019-3577(23)00080-0 DE-627 ger DE-627 rda eng 510 VZ 31.00 bkl Rodríguez, José verfasserin aut Weak precompactness in projective tensor products 2023 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . Projective tensor product Weakly compact set Weakly precompact set Coarse Rueda Zoca, Abraham verfasserin aut Enthalten in Indagationes mathematicae Amsterdam : Elsevier, 1990 35, Seite 60-75 Online-Ressource (DE-627)265549957 (DE-600)1465401-5 (DE-576)11281512X nnns volume:35 pages:60-75 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.00 Mathematik: Allgemeines VZ AR 35 60-75 |
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10.1016/j.indag.2023.08.003 doi (DE-627)ELV06643288X (ELSEVIER)S0019-3577(23)00080-0 DE-627 ger DE-627 rda eng 510 VZ 31.00 bkl Rodríguez, José verfasserin aut Weak precompactness in projective tensor products 2023 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . Projective tensor product Weakly compact set Weakly precompact set Coarse Rueda Zoca, Abraham verfasserin aut Enthalten in Indagationes mathematicae Amsterdam : Elsevier, 1990 35, Seite 60-75 Online-Ressource (DE-627)265549957 (DE-600)1465401-5 (DE-576)11281512X nnns volume:35 pages:60-75 GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2088 GBV_ILN_2106 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 31.00 Mathematik: Allgemeines VZ AR 35 60-75 |
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Weak precompactness in projective tensor products |
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Weak precompactness in projective tensor products |
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Rodríguez, José |
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Rodríguez, José Rueda Zoca, Abraham |
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weak precompactness in projective tensor products |
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Weak precompactness in projective tensor products |
abstract |
We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . |
abstractGer |
We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . |
abstract_unstemmed |
We give a sufficient condition for a pair of Banach spaces ( X , Y ) to have the following property: whenever W 1 ⊆ X and W 2 ⊆ Y are sets such that { x ⊗ y : x ∈ W 1 , y ∈ W 2 } is weakly precompact in the projective tensor product X ⊗ ̂ π Y , then either W 1 or W 2 is relatively norm compact. For instance, such a property holds for the pair ( ℓ p , ℓ q ) if 1 < p , q < ∞ satisfy 1 / p + 1 / q ≥ 1 . Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ 1 into X ⊗ ̂ π Y for arbitrary Banach spaces X and Y , in connection with the compactness of all operators from X to Y ∗ . |
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title_short |
Weak precompactness in projective tensor products |
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