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

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 ∗ . Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Rodríguez, José [verfasserIn] Rueda Zoca, Abraham [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2023

Schlagwörter:	Projective tensor product Weakly compact set Weakly precompact set Coarse

Übergeordnetes Werk:	Enthalten in: Indagationes mathematicae - Amsterdam : Elsevier, 1990, 35, Seite 60-75
Übergeordnetes Werk:	volume:35 ; pages:60-75

DOI / URN:	10.1016/j.indag.2023.08.003

Katalog-ID:	ELV06643288X

Internformat


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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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912			\|a GBV_ILN_370
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912			\|a GBV_ILN_4326
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912			\|a GBV_ILN_4367
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936	b	k	\|a 31.00 \|j Mathematik: Allgemeines \|q VZ
951			\|a AR
952			\|d 35 \|h 60-75

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author_variant	j r jr z a r za zar
matchkey_str	rodrguezjosruedazocaabraham:2023----:ekrcmatesnrjciee
hierarchy_sort_str	2023
bklnumber	31.00
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
allfieldsSound	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
language	English
source	Enthalten in Indagationes mathematicae 35, Seite 60-75 volume:35 pages:60-75
sourceStr	Enthalten in Indagationes mathematicae 35, Seite 60-75 volume:35 pages:60-75
format_phy_str_mv	Article
bklname	Mathematik: Allgemeines
institution	findex.gbv.de
topic_facet	Projective tensor product Weakly compact set Weakly precompact set Coarse
dewey-raw	510
isfreeaccess_bool	false
container_title	Indagationes mathematicae
authorswithroles_txt_mv	Rodríguez, José @@aut@@ Rueda Zoca, Abraham @@aut@@
publishDateDaySort_date	2023-01-01T00:00:00Z
hierarchy_top_id	265549957
dewey-sort	3510
id	ELV06643288X
language_de	englisch
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author	Rodríguez, José
spellingShingle	Rodríguez, José ddc 510 bkl 31.00 misc Projective tensor product misc Weakly compact set misc Weakly precompact set misc Coarse Weak precompactness in projective tensor products
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topic_title	510 VZ 31.00 bkl Weak precompactness in projective tensor products Projective tensor product Weakly compact set Weakly precompact set Coarse
topic	ddc 510 bkl 31.00 misc Projective tensor product misc Weakly compact set misc Weakly precompact set misc Coarse
topic_unstemmed	ddc 510 bkl 31.00 misc Projective tensor product misc Weakly compact set misc Weakly precompact set misc Coarse
topic_browse	ddc 510 bkl 31.00 misc Projective tensor product misc Weakly compact set misc Weakly precompact set misc Coarse
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title	Weak precompactness in projective tensor products
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title_full	Weak precompactness in projective tensor products
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title_sort	weak precompactness in projective tensor products
title_auth	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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up_date	2024-07-06T17:44:30.186Z
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Weak precompactness in projective tensor products

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