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).

Limiting diffusion for random walks with drift conditioned to stay positive

1978 ◽  
Vol 15 (02) ◽  
pp. 280-291 ◽  
Author(s):  
Peichuen Kao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Function ◽  
Random Variables ◽  
Principal Result ◽  
Hitting Time ◽  
Brownian Excursion ◽  
Norming Constant ◽  
Finite Dimensional ◽  
Independent Identically Distributed

Let {ξ k : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ 1} = μ ≠ 0. Form the random walk {S n : n ≧ 0} by setting S 0, S n = ξ 1 + ξ 2 + ··· + ξ n , n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ 1) that the finite-dimensional distributions of Xn , conditioned on n < N < ∞ converge to those of the Brownian excursion process.

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.

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.

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.

Almost transitivity of some function spaces

1994 ◽  
Vol 116 (3) ◽  
pp. 475-488 ◽  
Author(s):  
Peter Greim ◽  
James E. Jamison ◽  
Anna Kamińska
Keyword(s):  
Unit Ball ◽  
Orlicz Space ◽  
Orlicz Spaces ◽  
Extreme Points ◽  
Function Spaces ◽  
Separable Spaces

AbstractThe almost transitive norm problem is studied for Lp (μ, X), C(K, X) and for certain Orlicz and Musielak-Orlicz spaces. For example if p ≠ 2 < ∞ then Lp (μ) has almost transitive norm if and only if the measure μ is homogeneous. It is shown that the only Musielak-Orlicz space with almost transitive norm is the Lp-space. Furthermore, an Orlicz space has an almost transitive norm if and only if the norm is maximal. Lp (μ, X) has almost transitive norm if Lp(μ) and X have. Separable spaces with non-trivial Lp-structure fail to have transitive norms. Spaces with nontrivial centralizers and extreme points in the unit ball also fail to have almost transitive norms.

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