Operator ranges and quasicomplemented subspaces of Banach spaces

Fonf, V. P.; Lajara, Sebastián; Troyanski, Stanimir; Zanco, Clemente

doi:10.4064/sm180110-31-1

Cited by 5 publications

(2 citation statements)

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“…To prove this assertion we assume first that dim(E/Y ) = ∞. As n≥1 γ n K n is a symmetric compact convex subset of Y and R + ∩ Y ∈ S(Y ), thanks to [12,Lemma 3.3] we have R + + span ( n≥1 α n K n ) ∈ S(Y ). Thus, proceeding as in the proof of the claim (with n ≥ 2) we obtain a minimal sequence {x j } j ⊂ B E such that [{x j } j ] = E and the set K ω = co({±2 − j x j } j ) satisfies (a ω ).…”

Section: Lemma 52 Let {V M } M Be a Sequence Of Symmetric Closed Conv...mentioning

confidence: 99%

Operator ranges and endomorphisms with a prescribed behaviour on Banach spaces

Jiménez-Sevilla

Lajara

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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We obtain several extensions of a theorem of Shevchik which asserts that if R is a proper dense operator range in a separable Banach space E, then there exists a compact, one-to-one and dense-range operator $$T:E\rightarrow E$$ T : E → E such that $$T(E)\cap R = \{0\}$$ T ( E ) ∩ R = { 0 } , and some results of Chalendar and Partington concerning the existence of compact, one-to-one and dense-range endomorphisms on a separable Banach space E which leave invariant a given closed subspace $$Y\subset E$$ Y ⊂ E , or more generally, a countable increasing chain of closed subspaces of E.

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Section: Lemma 52 Let {V M } M Be a Sequence Of Symmetric Closed Conv...mentioning

confidence: 99%

Operator ranges and endomorphisms with a prescribed behaviour on Banach spaces

Jiménez-Sevilla

Lajara

2022

Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.

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“…Finally, there is a second sense in which the subspace Y p can be considered to be 'large', which is connected to the notion of operator ranges, [3,7,8]. Recall that a normed space Y is an operator range if there are a Banach space Z and a surjective bounded linear operator T : Z → Y; in other words, Y is the linear image of a Banach space.…”

Section: Introductionmentioning

confidence: 99%

Smooth norms in dense subspaces of $\ell_p(Γ)$ and operator ranges

Dantas¹,

Hájek²,

Russo³

2022

Preprint

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For 1 p < ∞, we prove that the dense subspace Y p of ℓ p (Γ) comprising all elements y such that y ∈ ℓ q (Γ) for some q ∈ (0, p) admits a C ∞ -smooth norm which locally depends on finitely many coordinates. Moreover, such a norm can be chosen as to approximate the • p -norm. This provides examples of dense subspaces of ℓ p (Γ) with a smooth norm which have the maximal possible linear dimension and are not obtained as the linear span of a biorthogonal system. Moreover, when p > 1 or Γ is countable, such subspaces additionally contain dense operator ranges; on the other hand, no non-separable operator range in ℓ 1 (Γ) admits a C 1 -smooth norm.

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Quasicomplemented Subspaces of Banach Spaces and Separable Quotients

Jiménez-Sevilla,

Lajara

2023

Results Math

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We present a result which generalizes the theorems of Gurariy-Kadets and Lindenstrauss-Rosenthal on quasicomplemented subspaces of Banach spaces. We also provide a solution in wide classes of Banach spaces of a problem posed by Ivan Singer concerning the existence of a $$w^*$$ w ∗ -basic sequence in the dual of a Banach space which is biorthogonal to a given basic sequence (or more generally, a given minimal sequence) in that space, and other related problems.

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Operator ranges and quasicomplemented subspaces of Banach spaces

Cited by 5 publications

References 0 publications

Operator ranges and endomorphisms with a prescribed behaviour on Banach spaces

Operator ranges and endomorphisms with a prescribed behaviour on Banach spaces

Smooth norms in dense subspaces of $\ell_p(Γ)$ and operator ranges

Quasicomplemented Subspaces of Banach Spaces and Separable Quotients

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