On the numerical index of vector-valued function spaces

Abstract

Let X be a Banach space and a positive measure. In this article, we show that nðL p ð, XÞÞ ¼ lim m nðl m p ðXÞÞ, 1 p < 1. Also, we investigate the positivity of the numerical index of l p -spaces.

On the numerical index of real L p (µ)-spaces

“…As M p > 0 for p = 2, this extends item (d) for infinite-dimensional real L p -spaces, meaning that the numerical radius and the operator norm are equivalent on L L p (µ) for every p = 2 and every positive measure µ. This answers in the positive a question raised by C. Finet and D. Li (see [5,6]) also posed in [7, Problem 1].…”

Numerical index of absolute sums of Banach spaces

“…Actually, we will give the result for vector-valued spaces. Let us say that all the results in this section are already known: they were proved using particular arguments of the L p spaces in the papers [7,8,9] (some of them with additional unnecessary hypotheses on the measure µ). In our opinion, the abstract vision we are developing in this paper allows to understand better the properties of L p -spaces underlying the proofs: ℓ p -sums are absolute sums, L p -norms are associative, every measure space can be decomposed into parts of finite measure, every finite measure algebra is isomorphic to the union of homogeneous measure algebras (Maharam's theorem) and, finally, the density of simple functions via the conditional expectation projections.…”

“…Since L p (µ) spaces are order continuous Köthe spaces for 1 p < ∞, as an immediate consequence of Corollary 4.2 we obtain the following corollary. For p = 1 it appeared in [19] and for 1 < p < ∞ it appeared in [9]. Proof.…”

“…If H is a Hilbert space of dimension greater than one then n(H) = 0 in the real case and n(H) = 1/2 in the complex case. The exact value of the numerical indices of L p (µ) spaces is still unknown when 1 < p < ∞ and p = 2, but it is is known [7,8] that all infinite-dimensional L p (µ) spaces have the same numerical index, which coincides with the infimum of the numerical indices of finite-dimensional L p (µ) spaces, and the result has been extended to vectorvalued L p spaces [9]. It has been shown very recently [18] that every real L p (µ) space has positive numerical index for p = 2.…”

Limit theorems for the numerical index

“…In the same paper it is also established that n(ℓ m p ) ̸ = 0 for finite m in the real case. In [6] the numerical index of vector-valued function spaces is considered and a proof of n(L p (µ, X )) = lim m n(ℓ m p (X))…”