Schauder-Orlicz decompositions, $\ell_Φ$-decompositions and pseudo-Daugavet property
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909106492669952 |
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| author | Marchenko, Vitalii |
| author_facet | Marchenko, Vitalii |
| contents | The concept of $\ell_Φ$-decomposition, extending the concept of $\ell_{p}$-decomposition of a Banach space, is presented and basic properties of Schauder-Orlicz decompositions and $\ell_Φ$-decompositions are studied. We show that Schauder-Orlicz decompositions are orthogonal in a sense of Grinblyum-James and Singer. Simple constructions of $\ell_{p}$-decompositions and Schauder-Orlicz decompositions in $L_p$ are presented. We prove that in the class of spaces possessing pseudo-Daugavet property, which includes classical $L_p$, $1\leq p\neq 2$, and $C$, Schauder-Orlicz decompositions with at least one finite dimensional subspace do not exist. It follows that Kato theorem on similarity for sequences of projections [1] cannot be extended to spaces from this class. Moreover we show that Banach spaces, possessing Schauder-Orlicz decompositions with at least one finite dimensional subspace, do not have pseudo-Daugavet property. Thus for Banach spaces $X$ possessing Schauder-Orlicz decompositions we obtain the following characterization of pseudo-Daugavet property: $X$ has pseudo-Daugavet property if and only if there is no Schauder-Orlicz decomposition in $X$ with at least one finite dimensional subspace if and only if there is no Schauder-Orlicz decomposition in $X$, which is an FDD. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09350 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schauder-Orlicz decompositions, $\ell_Φ$-decompositions and pseudo-Daugavet property Marchenko, Vitalii Functional Analysis 46B15, 46E30, 46B20, 46B50, 46B04, 46B45, 41A65 The concept of $\ell_Φ$-decomposition, extending the concept of $\ell_{p}$-decomposition of a Banach space, is presented and basic properties of Schauder-Orlicz decompositions and $\ell_Φ$-decompositions are studied. We show that Schauder-Orlicz decompositions are orthogonal in a sense of Grinblyum-James and Singer. Simple constructions of $\ell_{p}$-decompositions and Schauder-Orlicz decompositions in $L_p$ are presented. We prove that in the class of spaces possessing pseudo-Daugavet property, which includes classical $L_p$, $1\leq p\neq 2$, and $C$, Schauder-Orlicz decompositions with at least one finite dimensional subspace do not exist. It follows that Kato theorem on similarity for sequences of projections [1] cannot be extended to spaces from this class. Moreover we show that Banach spaces, possessing Schauder-Orlicz decompositions with at least one finite dimensional subspace, do not have pseudo-Daugavet property. Thus for Banach spaces $X$ possessing Schauder-Orlicz decompositions we obtain the following characterization of pseudo-Daugavet property: $X$ has pseudo-Daugavet property if and only if there is no Schauder-Orlicz decomposition in $X$ with at least one finite dimensional subspace if and only if there is no Schauder-Orlicz decomposition in $X$, which is an FDD. |
| title | Schauder-Orlicz decompositions, $\ell_Φ$-decompositions and pseudo-Daugavet property |
| topic | Functional Analysis 46B15, 46E30, 46B20, 46B50, 46B04, 46B45, 41A65 |
| url | https://arxiv.org/abs/2402.09350 |