The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
C. Leránoz - One of the best experts on this subject based on the ideXlab platform.
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Uniqueness of Unconditional bases in nonlocally convex c0-products
Israel Journal of Mathematics, 2011Co-Authors: Fernando Albiac, C. LeránozAbstract:We show that the c 0-product (X ⊕ X ⊕ ... ⊕ X ⊕ ...)0 of a natural quasi-Banach space X with strongly absolute Unconditional Basis has a unique Unconditional Basis up to permutation. Our results apply to a wide range of cases, including most of the c 0-products of the nonlocally convex classical quasi-Banach spaces.
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Uniqueness of Unconditional bases in nonlocally convex ℓ1-products☆
Journal of Mathematical Analysis and Applications, 2011Co-Authors: Fernando Albiac, C. LeránozAbstract:Abstract We show that the l 1 -product ( X ⊕ X ⊕ ⋯ ⊕ X ⊕ ⋯ ) 1 has a unique Unconditional Basis up to permutation for a wide class of nonlocally convex quasi-Banach spaces X, even without knowing whether X has a unique Unconditional Basis or not.
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Uniqueness of Unconditional Basis in quasi-Banach spaces which are not sufficiently Euclidean
Positivity, 2010Co-Authors: Fernando Albiac, C. LeránozAbstract:Strongly absolute bases are, roughly speaking, purely nonlocally convex bases in quasi-Banach spaces. When, in addition, they are Unconditional then the discrete lattice structure they induce in the space is lattice anti-Euclidean. In this brief note we characterize the complemented Unconditional basic sequences in those quasi-Banach spaces with strongly absolute Unconditional Basis, and use this result to derive the uniqueness of Unconditional Basis in many classical quasi-Banach spaces.
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An alternative approach to the uniqueness of Unconditional Basis of ℓp(c0) for 0
Expositiones Mathematicae, 2010Co-Authors: Fernando Albiac, C. LeránozAbstract:Abstract We give an alternative and much simpler proof of the uniqueness of Unconditional Basis (up to equivalence and permutation) in the quasi-Banach spaces l p ( c 0 ) for 0 p 1 and its complemented subspaces with Unconditional Basis. The new approach uses the fact that the Banach envelope of these spaces is not sufficiently Euclidean with the lattice structure induced by its Unconditional Basis.
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an alternative approach to the uniqueness of Unconditional Basis of lp c0 for 0 p 1
Expositiones Mathematicae, 2010Co-Authors: Fernando Albiac, C. LeránozAbstract:Abstract We give an alternative and much simpler proof of the uniqueness of Unconditional Basis (up to equivalence and permutation) in the quasi-Banach spaces l p ( c 0 ) for 0 p 1 and its complemented subspaces with Unconditional Basis. The new approach uses the fact that the Banach envelope of these spaces is not sufficiently Euclidean with the lattice structure induced by its Unconditional Basis.
Fernando Albiac - One of the best experts on this subject based on the ideXlab platform.
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uniqueness of Unconditional Basis of infinite direct sums of quasi banach spaces
arXiv: Functional Analysis, 2021Co-Authors: Fernando Albiac, Jose L. AnsorenaAbstract:This paper is devoted to providing a unifying approach to the study of the uniqueness of Unconditional bases, up to equivalence and permutation, of infinite direct sums of quasi-Banach spaces. Our new approach to this type of problem permits us to show that a wide class of vector-valued sequence spaces have a unique Unconditional Basis up to a permutation. In particular, solving a problem from [F. Albiac and C. Ler\'anoz, Uniqueness of Unconditional bases in nonlocally convex $\ell_1$-products, J. Math. Anal. Appl. 374 (2011), no. 2, 394--401] we show that if $X$ is quasi-Banach space with a strongly absolute Unconditional Basis then the infinite direct sum $\ell_{1}(X)$ has a unique Unconditional Basis up to a permutation, even without knowing whether $X$ has a unique Unconditional Basis or not. Applications to the uniqueness of Unconditional structure of infinite direct sums of non-locally convex Orlicz and Lorentz sequence spaces, among other classical spaces, are also obtained as a by-product of our work.
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uniqueness of Unconditional Basis of ell_ 2 oplus mathcal t 2
arXiv: Functional Analysis, 2020Co-Authors: Fernando Albiac, Jose L. AnsorenaAbstract:We provide a new extension of Pitt's theorem for compact operators between quasi-Banach lattices, which permits to describe Unconditional bases of finite direct sums of Banach spaces $\mathbb{X}_{1}\oplus\dots\oplus\mathbb{X}_{n}$ as direct sums of Unconditional bases of its summands. The general splitting principle we obtain yields, in particular, that if each $\mathbb{X}_{i}$ has a unique Unconditional Basis (up to equivalence and permutation), then $\mathbb{X}_{1}\oplus \cdots\oplus\mathbb{X}_{n}$ has a unique Unconditional Basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space $\ell_2\oplus \mathcal{T}^{(2)}$ has a unique Unconditional Basis.
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Uniqueness of Unconditional Basis of $\ell_{2}\oplus \mathcal{T}^{(2)}$
arXiv: Functional Analysis, 2020Co-Authors: Fernando Albiac, Jose L. AnsorenaAbstract:We provide a new extension of Pitt's theorem for compact operators between quasi-Banach lattices, which permits to describe Unconditional bases of finite direct sums of Banach spaces $\mathbb{X}_{1}\oplus\dots\oplus\mathbb{X}_{n}$ as direct sums of Unconditional bases of its summands. The general splitting principle we obtain yields, in particular, that if each $\mathbb{X}_{i}$ has a unique Unconditional Basis (up to equivalence and permutation), then $\mathbb{X}_{1}\oplus \cdots\oplus\mathbb{X}_{n}$ has a unique Unconditional Basis too. Among the novel applications of our techniques to the structure of Banach and quasi-Banach spaces we have that the space $\ell_2\oplus \mathcal{T}^{(2)}$ has a unique Unconditional Basis.
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Uniqueness of Unconditional Basis of $H_p(\mathbb{T})\oplus\ell_{2}$ and $H_p(\mathbb{T})\oplus\mathcal{T}^{(2)}$ for $0
arXiv: Functional Analysis, 2020Co-Authors: Fernando Albiac, Jose L. Ansorena, Przemysław WojtaszczykAbstract:Our goal in this paper is to advance the state of the art of the topic of uniqueness of Unconditional Basis. To that end we establish general conditions on a pair $(\mathbb{X}, \mathbb{Y})$ formed by a quasi-Banach space $\mathbb{X}$ and a Banach space $\mathbb{Y}$ which guarantee that every Unconditional Basis of their direct sum $\mathbb{X}\oplus\mathbb{Y}$ splits into Unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces $H_p(\mathbb{T}^d) \oplus\mathcal{T}^{(2)}$ and $H_p(\mathbb{T}^d)\oplus\ell_2$, for $p\in(0,1)$ and $d\in\mathbb{N}$, have a unique Unconditional Basis (up to equivalence and permutation).
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uniqueness of Unconditional Basis of h_p mathbb t oplus ell_ 2 and h_p mathbb t oplus mathcal t 2 for 0 p 1
arXiv: Functional Analysis, 2020Co-Authors: Fernando Albiac, Jose L. Ansorena, Przemysław WojtaszczykAbstract:Our goal in this paper is to advance the state of the art of the topic of uniqueness of Unconditional Basis. To that end we establish general conditions on a pair $(\mathbb{X}, \mathbb{Y})$ formed by a quasi-Banach space $\mathbb{X}$ and a Banach space $\mathbb{Y}$ which guarantee that every Unconditional Basis of their direct sum $\mathbb{X}\oplus\mathbb{Y}$ splits into Unconditional bases of each summand. As application of our methods we obtain that, among others, the spaces $H_p(\mathbb{T}^d) \oplus\mathcal{T}^{(2)}$ and $H_p(\mathbb{T}^d)\oplus\ell_2$, for $p\in(0,1)$ and $d\in\mathbb{N}$, have a unique Unconditional Basis (up to equivalence and permutation).
Ioannis Gasparis - One of the best experts on this subject based on the ideXlab platform.
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New examples of c0-saturated Banach spaces II
Journal of Functional Analysis, 2009Co-Authors: Ioannis GasparisAbstract:Abstract For every Banach space Z with a shrinking Unconditional Basis satisfying an upper p -estimate for some p > 1 , an isomorphically polyhedral Banach space is constructed which has an Unconditional Basis and admits a quotient isomorphic to Z . It follows that reflexive Banach spaces with an Unconditional Basis and non-trivial type, Tsirelson's original space and ( ∑ c 0 ) l p for p ∈ ( 1 , ∞ ) , are isomorphic to quotients of isomorphically polyhedral Banach spaces with Unconditional bases.
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New examples of $c_0$-saturated Banach spaces II
arXiv: Functional Analysis, 2008Co-Authors: Ioannis GasparisAbstract:For every Banach space $Z$ with a shrinking Unconditional Basis satisfying upper $p$-estimates for some $p > 1$, an isomorphically polyhedral Banach space is constructed having an Unconditional Basis and admitting a quotient isomorphic to $Z$.
Manuel Maestre - One of the best experts on this subject based on the ideXlab platform.
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Existence of Unconditional Bases in Spaces of Polynomials and Holomorphic Functions
Mathematische Nachrichten, 2002Co-Authors: Andreas Defant, Juan Carlos Díaz, Domingo García, Manuel MaestreAbstract:Our main result shows that every Montel Kothe echelon or coechelon space E of order 1 < p ≤ ∞ is nuclear if and only if for every (some) m ≥ 2 the space ((mE), τ0) of m-homegeneus polynomials on E endowed with the compact-open topology τ0 has an Unconditional Basis if and only if the space (ℋ(E), τδ) of holomorphic functions on E endowed with the bornological topology τδ associated to τ0 has an Unconditional Basis (for coechelon spaces τδ equals τ0). The main idea is to extend the concept of the Gordon-Lewis property from Banach to Frechet and (DF) spaces. In this way we obtain techniques which are used to characterize the existence of Unconditional Basis in spaces of m-th (symmetric) tensor products and, as a consequence, in spaces of polynomials and holomorphic functions.
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Unconditional Basis and gordon lewis constants for spaces of polynomials
Journal of Functional Analysis, 2001Co-Authors: Andreas Defant, Juan Carlos Díaz, Domingo García, Manuel MaestreAbstract:Abstract No infinite dimensional Banach space X is known which has the property that for m ⩾2 the Banach space of all continuous m -homogeneous polynomials on X has an Unconditional Basis. Following a program originally initiated by Gordon and Lewis we study Unconditionality in spaces of m -homogeneous polynomials and symmetric tensor products of order m in Banach spaces. We show that for each Banach space X which has a dual with an Unconditional Basis ( x * i ), the approximable (nuclear) m -homogeneous polynomials on X have an Unconditional Basis if and only if the monomial Basis with respect to ( x * i ) is Unconditional. Moreover, we determine an asymptotically correct estimate for the Unconditional Basis constant of all m -homogeneous polynomials on l n p and use this result to narrow down considerably the list of natural candidates X with the above property.
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Unconditional Basis and Gordon–Lewis Constants for Spaces of Polynomials
Journal of Functional Analysis, 2001Co-Authors: Andreas Defant, Juan Carlos Díaz, Domingo García, Manuel MaestreAbstract:Abstract No infinite dimensional Banach space X is known which has the property that for m ⩾2 the Banach space of all continuous m -homogeneous polynomials on X has an Unconditional Basis. Following a program originally initiated by Gordon and Lewis we study Unconditionality in spaces of m -homogeneous polynomials and symmetric tensor products of order m in Banach spaces. We show that for each Banach space X which has a dual with an Unconditional Basis ( x * i ), the approximable (nuclear) m -homogeneous polynomials on X have an Unconditional Basis if and only if the monomial Basis with respect to ( x * i ) is Unconditional. Moreover, we determine an asymptotically correct estimate for the Unconditional Basis constant of all m -homogeneous polynomials on l n p and use this result to narrow down considerably the list of natural candidates X with the above property.
Nigel J. Kalton - One of the best experts on this subject based on the ideXlab platform.
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Unconditionality in spaces of m-homogeneous polynomials
The Quarterly Journal of Mathematics, 2005Co-Authors: Andreas Defant, Nigel J. KaltonAbstract:Let E be a Banach space with an Unconditional Basis. We prove that for m 2 the Banach space P( m E) of all m-homogeneous polynomials on E has an Unconditional Basis if and only if E is finite dimensional. This answers a problem of S. Dineen.
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uniqueness of the Unconditional Basis of l 1 l p and l p l 1 0 p 1
Positivity, 2004Co-Authors: Fernando Albiac, Nigel J. Kalton, C. LeránozAbstract:We prove that the quasi-Banach spaces l1 (lp) and lp (l1), 0 < p < 1 have a unique Unconditional Basis up to permutation
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Uniqueness of the Unconditional Basis of ℓ 1 (ℓ p ) and ℓ p (ℓ 1 ), 0 < p < 1
Positivity, 2004Co-Authors: Fernando Albiac, Nigel J. Kalton, C. LeránozAbstract:We prove that the quasi-Banach spaces l1 (lp) and lp (l1), 0 < p < 1 have a unique Unconditional Basis up to permutation
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TWISTED HILBERT SPACES AND Unconditional STRUCTURE
Journal of The Institute of Mathematics of Jussieu, 2003Co-Authors: Nigel J. KaltonAbstract:We show that a twisted Hilbert space with an Unconditional Basis is isomorphic to a Hilbert space.
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INTERPOLATION OF SUBSPACES AND APPLICATIONS TO EXPONENTIAL BASES IN SOBOLEV SPACES
arXiv: Functional Analysis, 2001Co-Authors: S. A. Ivanov, Nigel J. KaltonAbstract:We give precise conditions under which the real interpolation space (Y0,X1)�,p coincides with a closed subspace of (X0,X1)�,p when Y0 is a closed subspace of codimension one. We then apply this result to nonharmonic Fourier series in Sobolev spaces H s (−�,�) when 0 < s < 1. The main result: let E be a family of exponentials exp(int) and E forms an Unconditional Basis in L 2 (−�,�). Then there exist two number s0,s1 such that E forms an Unconditional Basis in H s for s < s0, E forms an Unconditional Basis in its span with codimension 1 in H s for s1 < s. For s0 ≤ s ≤ s1 the exponential family is not an Unconditional Basis in its span.
