scholarly journals About interpolation of subspaces of rearrangement invariant spaces generated by Rademacher system

2001 ◽  
Vol 25 (7) ◽  
pp. 451-465 ◽  
Author(s):  
Sergey V. Astashkin
Keyword(s):  
Invariant Function ◽  
Function Spaces ◽  
Interpolation Theory ◽  
Rearrangement Invariant Spaces ◽  
Rademacher System ◽  
Rearrangement Invariant Function Spaces ◽  
One To One ◽  
Rademacher Series ◽  
Invariant Spaces

The Rademacher series in rearrangement invariant function spaces “close” to the spaceL∞are considered. In terms of interpolation theory of operators, a correspondence between such spaces and spaces of coefficients generated by them is stated. It is proved that this correspondence is one-to-one. Some examples and applications are presented.

scholarly journals Isometries of Hilbert space valued function spaces

1996 ◽  
Vol 61 (2) ◽  
pp. 150-161 ◽  
Author(s):  
Beata Randrianantoanina
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Invariant Function ◽  
Function Spaces ◽  
Measurable Operator ◽  
Complex Rearrangement ◽  
Rearrangement Invariant Function Spaces ◽  
Surjective Isometry ◽  
Almost All

AbstractLet X be a (real or complex) rearrangement-invariant function space on Ω (where Ω = [0, 1] or Ω ⊆ N) whose norm is not proportional to the L2-norm. Let H be a separable Hilbert space. We characterize surjective isometries of X (H). We prove that if T is such an isometry then there exist Borel maps a: Ω → + K and σ: Ω → Ω and a strongly measurable operator map S of Ω into B (H) so that for almost all ω, S(ω) is a surjective isometry of H, and for any f ∈ X(H), T f(ω) = a(ω)S(ω)(f(σ(ω))) a.e. As a consequence we obtain a new proof of the characterization of surjective isometries in complex rearrangement-invariant function spaces.

A note on symmetric basic sequences in Lp(Lq)

1992 ◽  
Vol 112 (1) ◽  
pp. 183-194
Author(s):  
Yves Raynaud
Keyword(s):  
Orlicz Space ◽  
Orlicz Spaces ◽  
Random Variables ◽  
Invariant Function ◽  
Function Spaces ◽  
Symmetric Basis ◽  
Finite Dimensional ◽  
Rearrangement Invariant Function Spaces ◽  
Independent Identically Distributed

Subspaces of Lp spanned by symmetric independent identically distributed random variables were identified as Orlicz spaces by Bretagnolle and Dacunha-Castelle[1], who showed that, conversely, in the case p ≤ 2, every p-convex, 2-concave Orlicz space is isomorphic to a subspace of Lp. This was extended by Dacunha-Castelle [3] to subspaces of Lp with symmetric basis, which appear as ‘p-means’ of Orlicz spaces (see [9] for the corresponding finite-dimensional result, and [12] for the case of rearrangement invariant function spaces). On the contrary the only subspaces with symmetric basis of Lp for p ≥ 2 are lp and l2 (if one does not care about isomorphy constants).

scholarly journals Weak-Type Boundedness of the Fourier Transform on Rearrangement Invariant Function Spaces

2018 ◽  
Vol 61 (3) ◽  
pp. 879-890
Author(s):  
Santiago Boza ◽  
Javier Soria
Keyword(s):  
Fourier Transform ◽  
Weak Type ◽  
Bounded Operator ◽  
Invariant Function ◽  
Rearrangement Invariant Space ◽  
Rearrangement Invariant Function Spaces ◽  
Weighted Lorentz Spaces ◽  
The Fourier Transform ◽  
Invariant Spaces ◽  
Maximal Space

AbstractWe study several questions about the weak-type boundedness of the Fourier transform ℱ on rearrangement invariant spaces. In particular, we characterize the action of ℱ as a bounded operator from the minimal Lorentz space Λ(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of ℱ, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.

Disjointly strictly singular operators and interpolation

1996 ◽  
Vol 126 (5) ◽  
pp. 1011-1026 ◽  
Author(s):  
A. García del Amo ◽  
F. L. Hernández ◽  
C. Ruiz
Keyword(s):  
Lattice Structure ◽  
Orlicz Function ◽  
Invariant Function ◽  
Function Spaces ◽  
Banach Lattices ◽  
Singular Operators ◽  
Rearrangement Invariant Function Spaces ◽  
Strictly Singular

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.

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