The Experts below are selected from a list of 87 Experts worldwide ranked by ideXlab platform
Serguei V Astashkin - One of the best experts on this subject based on the ideXlab platform.
-
Disjointly homogeneous Orlicz spaces revisited
Annali di Matematica Pura ed Applicata (1923 -), 2021Co-Authors: Serguei V AstashkinAbstract:Let $$1 \le p \le \infty $$ 1 ≤ p ≤ ∞ . A Banach lattice X is said to be p -Disjointly homogeneous or $$(p-DH)$$ ( p - D H ) (resp. restricted $$(p-DH)$$ ( p - D H ) ) if every normalized Disjoint Sequence in X (resp. every normalized Sequence of characteristic functions of Disjoint subsets) contains a subSequence equivalent in X to the unit vector basis of $$\ell _p$$ ℓ p . We revisit DH -properties of Orlicz spaces and refine some previous results of this topic, showing that the $$(p-DH)$$ ( p - D H ) -property is not stable under duality in the class of Orlicz spaces and the classes of restricted $$(p-DH)$$ ( p - D H ) and $$(p-DH)$$ ( p - D H ) Orlicz spaces are different. Moreover, we give a characterization of uniform $$(p-DH)$$ ( p - D H ) Orlicz spaces and establish also closed connections between this property and the duality of the DH -property.
-
Disjointly homogeneous Orlicz spaces revisited
arXiv: Functional Analysis, 2020Co-Authors: Serguei V AstashkinAbstract:Let $1\le p\le\infty$. A Banach lattice $X$ is said to be $p$-Disjointly homogeneous or $(p-DH)$ (resp. restricted $(p-DH)$) if every normalized Disjoint Sequence in $X$ (resp. every normalized Sequence of characteristic functions of Disjoint subsets) contains a subSequence equivalent in $X$ to the unit vector basis of $\ell_p$. We revisit $DH$-properties of Orlicz spaces and refine some previous results of this topic, showing that $(p-DH)$-property is not stable in the class of Orlicz spaces and the classes of restricted $(p-DH)$ and $(p-DH)$ Orlicz spaces are different. Moreover, we give a characterization of uniform $(p-DH)$ Orlicz spaces and establish also closed connections between this property and the duality of $DH$-property.
-
Duality problem for Disjointly homogeneous rearrangement invariant spaces
Journal of Functional Analysis, 2019Co-Authors: Serguei V AstashkinAbstract:Abstract Let 1 ≤ p ∞ . A Banach lattice E is said to be Disjointly homogeneous (resp. p-Disjointly homogeneous) if two arbitrary normalized Disjoint Sequences from E contain equivalent in E subSequences (resp. every normalized Disjoint Sequence contains a subSequence equivalent in E to the unit vector basis of l p ). Answering a question raised in the paper [11] , for each 1 p ∞ , we construct a reflexive p-Disjointly homogeneous rearrangement invariant space on [ 0 , 1 ] whose dual is not Disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of Disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type Disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to l p , 1 ≤ p ∞ , or c 0 .
-
Duality problem for Disjointly homogeneous rearrangement invariant spaces
Journal of Functional Analysis, 2019Co-Authors: Serguei V AstashkinAbstract:Abstract Let 1 ≤ p ∞ . A Banach lattice E is said to be Disjointly homogeneous (resp. p-Disjointly homogeneous) if two arbitrary normalized Disjoint Sequences from E contain equivalent in E subSequences (resp. every normalized Disjoint Sequence contains a subSequence equivalent in E to the unit vector basis of l p ). Answering a question raised in the paper [11] , for each 1 p ∞ , we construct a reflexive p-Disjointly homogeneous rearrangement invariant space on [ 0 , 1 ] whose dual is not Disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of Disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type Disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to l p , 1 ≤ p ∞ , or c 0 .
Pedro Tradacete - One of the best experts on this subject based on the ideXlab platform.
-
Strict Singularity: A Lattice Approach
Trends in Mathematics, 2019Co-Authors: Julio Flores, Francisco L. Hernández, Pedro TradaceteAbstract:Given a Banach lattice E and a Banach space Y we say that a bounded linear operator T : E → Y is lattice strictly singular (Disjointly strictly singular) if it fails to be invertible on any infinite-dimensional sublattice of E (on the span of any pairwise Disjoint Sequence in E). This is a survey on the existing answers up to the present day to the following questions: Is every lattice strictly singular operator also Disjointly strictly singular? Do lattice strictly singular operators have a vector space structure?
-
banach lattice versions of strict singularity
Journal of Functional Analysis, 2016Co-Authors: Julio Flores, Jordi Lopezabad, Pedro TradaceteAbstract:Abstract We explore the relation between lattice versions of strict singularity for operators from a Banach lattice to a Banach space. In particular, we study when the class of Disjointly strictly singular operators, those not invertible on the span of any Disjoint Sequence, coincides with that of lattice strictly singular operators, i.e. those not invertible on any (infinite dimensional) sublattice. New results are given which help to clarify the existing relation between these two classes.
-
Rearrangement invariant spaces with Kato property
Functiones et Approximatio Commentarii Mathematici, 2014Co-Authors: Francisco L. Hernández, Evgueni M. Semenov, Pedro TradaceteAbstract:We study rearrangement invariant spaces on which the classes of strictly singular and compact operators coincide. The relation between this property and the fact that every normalized Disjoint Sequence in the space has a subSequence equivalent to the unit vector basis of $\ell_2$ is analyzed.
Evgueni M. Semenov - One of the best experts on this subject based on the ideXlab platform.
-
Rearrangement invariant spaces with Kato property
Functiones et Approximatio Commentarii Mathematici, 2014Co-Authors: Francisco L. Hernández, Evgueni M. Semenov, Pedro TradaceteAbstract:We study rearrangement invariant spaces on which the classes of strictly singular and compact operators coincide. The relation between this property and the fact that every normalized Disjoint Sequence in the space has a subSequence equivalent to the unit vector basis of $\ell_2$ is analyzed.
-
DisjointLY STRICTLY-SINGULAR INCLUSIONS BETWEEN REARRANGEMENT INVARIANT SPACES
Journal of the London Mathematical Society, 2000Co-Authors: A. García Del Amo, Francisco L. Hernández, Víctor M. Sánchez, Evgueni M. SemenovAbstract:A linear operator between two Banach spaces X and Y is strictly-singular (or Kato) if it fails to be an isomorphism on any infinite dimensional subspace. A weaker notion for Banach lattices introduced in [8] is the following one: an operator T from a Banach lattice X to a Banach space Y is said to be Disjointly strictly-singular if there is no Disjoint Sequence of non-null vectors (xn)n∈N in X such that the restriction of T to the subspace [(xn)∞n=1] spanned by the vectors (xn)n∈N is an isomorphism. Clearly every strictly-singular operator is Disjointly strictly-singular but the converse is not true in general (consider for example the canonic inclusion Lq[0, 1]↪Lp[0, 1] for 1≤p
Francisco L. Hernández - One of the best experts on this subject based on the ideXlab platform.
-
Strict Singularity: A Lattice Approach
Trends in Mathematics, 2019Co-Authors: Julio Flores, Francisco L. Hernández, Pedro TradaceteAbstract:Given a Banach lattice E and a Banach space Y we say that a bounded linear operator T : E → Y is lattice strictly singular (Disjointly strictly singular) if it fails to be invertible on any infinite-dimensional sublattice of E (on the span of any pairwise Disjoint Sequence in E). This is a survey on the existing answers up to the present day to the following questions: Is every lattice strictly singular operator also Disjointly strictly singular? Do lattice strictly singular operators have a vector space structure?
-
Rearrangement invariant spaces with Kato property
Functiones et Approximatio Commentarii Mathematici, 2014Co-Authors: Francisco L. Hernández, Evgueni M. Semenov, Pedro TradaceteAbstract:We study rearrangement invariant spaces on which the classes of strictly singular and compact operators coincide. The relation between this property and the fact that every normalized Disjoint Sequence in the space has a subSequence equivalent to the unit vector basis of $\ell_2$ is analyzed.
-
DisjointLY STRICTLY-SINGULAR INCLUSIONS BETWEEN REARRANGEMENT INVARIANT SPACES
Journal of the London Mathematical Society, 2000Co-Authors: A. García Del Amo, Francisco L. Hernández, Víctor M. Sánchez, Evgueni M. SemenovAbstract:A linear operator between two Banach spaces X and Y is strictly-singular (or Kato) if it fails to be an isomorphism on any infinite dimensional subspace. A weaker notion for Banach lattices introduced in [8] is the following one: an operator T from a Banach lattice X to a Banach space Y is said to be Disjointly strictly-singular if there is no Disjoint Sequence of non-null vectors (xn)n∈N in X such that the restriction of T to the subspace [(xn)∞n=1] spanned by the vectors (xn)n∈N is an isomorphism. Clearly every strictly-singular operator is Disjointly strictly-singular but the converse is not true in general (consider for example the canonic inclusion Lq[0, 1]↪Lp[0, 1] for 1≤p
Julio Flores - One of the best experts on this subject based on the ideXlab platform.
-
Strict Singularity: A Lattice Approach
Trends in Mathematics, 2019Co-Authors: Julio Flores, Francisco L. Hernández, Pedro TradaceteAbstract:Given a Banach lattice E and a Banach space Y we say that a bounded linear operator T : E → Y is lattice strictly singular (Disjointly strictly singular) if it fails to be invertible on any infinite-dimensional sublattice of E (on the span of any pairwise Disjoint Sequence in E). This is a survey on the existing answers up to the present day to the following questions: Is every lattice strictly singular operator also Disjointly strictly singular? Do lattice strictly singular operators have a vector space structure?
-
banach lattice versions of strict singularity
Journal of Functional Analysis, 2016Co-Authors: Julio Flores, Jordi Lopezabad, Pedro TradaceteAbstract:Abstract We explore the relation between lattice versions of strict singularity for operators from a Banach lattice to a Banach space. In particular, we study when the class of Disjointly strictly singular operators, those not invertible on the span of any Disjoint Sequence, coincides with that of lattice strictly singular operators, i.e. those not invertible on any (infinite dimensional) sublattice. New results are given which help to clarify the existing relation between these two classes.