An estimate for narrow operators on $$L^p([0, 1])$$ L p ( [ 0 , 1 ] )
- PDF / 304,927 Bytes
- 6 Pages / 439.37 x 666.142 pts Page_size
- 27 Downloads / 243 Views
Archiv der Mathematik
An estimate for narrow operators on Lp ([0, 1]) Eugene Shargorodsky and Teo Sharia
Abstract. We prove a theorem, which generalises C. Franchetti’s estimate for the norm of a projection onto a rich subspace of Lp ([0, 1]) and the authors’ related estimate for compact operators on Lp ([0, 1]), 1 ≤ p < ∞. Mathematics Subject Classification. 47A30, 47B07, 47B38, 46E30. Keywords. Narrow operator, Norm, Estimate.
1. Introduction. For Banach spaces X and Y , let B(X, Y ) and K(X, Y ) denote the sets of bounded linear and compact linear operators from X to Y , respectively; B(X) := B(X, X), K(X) := K(X, X); I ∈ B(X) denotes the identity operator. An operator P ∈ B(X) is called a projection if P 2 = P . A closed linear subspace X0 ⊂ X is called 1-complemented (in X) if there exists a projection P ∈ B(X) such that P (X) = X0 and P = 1. Let (Ω, Σ, μ) be a nonatomic measure space with 0 < μ(Ω) < ∞. We will use the following notation: • Σ+ := {A ∈ Σ : μ(A) > 0}, • IA is the indicator function of A ∈ Σ, i.e. IA (ω) = 1 if ω ∈ A and IA (ω) = 0 if ω ∈ A, • 1 := IΩ, 1 • Ef := μ(Ω) f dμ 1. Ω We will use the terminology from [6]. A Σ-measurable function g is called a sign if it takes values in the set {−1, 0, 1}, and a sign on A ∈ Σ if it is a 2 sign with the support equal to A, i.e. if g = IA . A sign is of mean zero if g dμ = 0. Ω An operator T ∈ B(Lp (μ), Y ), 1 ≤ p < ∞, is called narrow if for every A ∈ Σ+ and every ε > 0, there exists a mean zero sign g on A such that T g < ε.
E. Shargorodsky and T. Sharia
Arch. Math.
Every T ∈ K(Lp (μ), Y ) is narrow (see [6, Proposition 2.1]), but there are noncompact narrow operators. Indeed, let G be a sub-σ-algebra of Σ such 1 μ , which is indepenthat there exists a random variable ξ on Ω, Σ, μ(Ω) dent of G and has a nontrivial Gaussian distribution. Then the corresponding conditional expectation operator EG = E(·|G) ∈ B(Lp (μ)) is narrow (see [6, Corollary 4.25]), but not compact if G has infinitely many pairwise disjoint elements of positive measure. Let 1− p1 1 1 1 Cp := max αp−1 + (1 − α)p−1 p α p−1 + (1 − α) p−1 (1.1) 0≤α≤1
for 1 < p < ∞, and C1 := 2. In the following theorems, (Ω, Σ, μ) = ([0, 1], L, λ), where λ is the standard Lebesgue measure on [0, 1] and L is the σ-algebra of Lebesgue measurable subsets of [0, 1]. Our starting point is a result due to C. Franchetti. Theorem 1.1 ([3,4]). Let P ∈ B(Lp ([0, 1]))\{0} be a narrow projection operator, 1 ≤ p < ∞. Then I − P Lp →Lp ≥ I − ELp →Lp = Cp .
(1.2)
The following theorem was proved in [7], where it was used to show that Cp is the optimal constant in the bounded compact approximation property of Lp ([0, 1]). It implies the inequality in (1.2) in the case when P = 0 is a finite-rank projection. Theorem 1.2 ([7]). Let 1 ≤ p < ∞, γ ∈ C, and let T ∈ K(Lp ([0, 1])). Then I − T Lp →Lp +
inf
uLp =1
(γI − T )uLp ≥ I − γELp →Lp .
(1.3)
In particular, I − T Lp →Lp +
inf
uLp =1
(I − T )uLp ≥ Cp .
(1.4)
The following theorem is the main result of the paper. It gen
Data Loading...
