On the numerical index of Banach spaces

“…The following result is well known and follows from the fact (see [2], §9, and [12]) that n(L p ) = 1 for p = 1, ∞, and the fact that self-adjoint operators have numericalradius equal to the operator-norm. We state this fact as a theorem (and include a proof for the sake of completeness), so that we can mention several corollaries, which are the results of recent work, and so that we can apply this theorem in Example 4.2 after Theorem 4.1 below.…”

Section: Diagonal Extremal Pairs and Minimal Projections In Lmentioning

confidence: 91%

Minimal numerical-radius extensions of operators

Aksoy

Chalmers

2007

Proc. Amer. Math. Soc.

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Abstract. In this paper we characterize minimal numerical-radius extensions of operators from finite-dimensional subspaces and compare them with minimal operator-norm extensions. We note that in the cases L p , p = 1, ∞, and in the case of self-adjoint extensions in L 2 , the two extensions and their norms are equal.We also show that, in the case of L p , 1 < p < ∞, and more generally in the case of the dual space being strictly convex, if the minimal projections with respect to the operator norm and with respect to the numerical radius have equal norms, then the operator norm is 1. An analogous result is also true for an arbitrary extension. Finally, we provide an example of a projection from l p 3 onto a two-dimensional subspace which is minimal with respect to norm but not with respect to the numerical radius for p = 1, 2, ∞, and we determine the minimal numerical-radius projection in this same situation.

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“…Actually, we prove that for 1 ≤ p < ∞, the numerical index of the Banach space L p ([0, 1], µ), where µ is the Lebesgue measure on the unit interval, is equal to the numerical index of the l p space. It is also known that the numerical index of the Banach space l p is the limit of the sequence of numerical index of finite-dimensional subspaces l m p , m = 1, 2, ..., [6]. The computation of the numerical index of the l m p -space then gives a complete answer to the problem of the numerical index of the L p -space.…”

Section: Clearly V Is a Semi-norm On B(x) And V(t ) ≤ T For Every Tmentioning

confidence: 99%