Numerical Radius of Rank-1 Operators on Banach Spaces

Abstract

We study the rank-one numerical index of a Banach space, namely the inmum of the numerical radii of those rank-one operators on the space which have norm-one. We show that the rank-one numerical index is always greater or equal than 1/ e. We also present properties of this index and some examples.

The Bishop–Phelps–Bollobás Property and Absolute Sums

“…We will profusely use in this section the following result which is a particular case of [14,Lemma 3.3]. We are now able to provide the proof of Theorem 4.1.…”

Rank-1 numerical index of some families of norms on the plane

“…We claim that T γ = (3−2γ ) 2 2−γ and v γ (T ) = 3−2γ 2−γ . Indeed, using the fact that 1 2 ≤ γ < 1 and Equations (1) and (2) one obtains…”

“…This concept was introduced in [1] as an analogue to the numerical index of Banach spaces and some of its properties were studied very recently in [2]. Let us recall the relevant definitions.…”

“…It is proved in [2] that n 1 (X ) ≥ 1/ e for every real Banach space X . It is also shown there that the rank-1 numerical index behaves similarly to the usual numerical index in some aspects.…”

“…It is also shown there that the rank-1 numerical index behaves similarly to the usual numerical index in some aspects. Namely, it is proved in [2] that the rank-1 numerical index of a c 0 , 1 and ∞ sum of spaces is the infimum of the rank-1 numerical index of the summands and the rank-1 numerical index of C(K , X ), L 1 (μ, X ) and L ∞ (μ, X ) is calculated. Besides, it is shown that the rank-1 numerical index is continuous with respect to equivalent norms.…”