Minimal numerical-radius extensions of operators

Abstract

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.

On the $L_p$ norm of the Rademacher projection and related inequalities

“…Interestingly, because of the form of the norming points of P, we can see that it is not a minimal projection with respect to the numerical radius (see [1] for details). As a result of the above theorem, we automatically get Theorem 2.11.…”

Best Approximation in Numerical Radius

“…and consequently f (a 1 bx + (1 − a 1 )y) = 0 Since f (x) = 0 and f (y) = 0, we can find exactly one c 1 …”

“…By [1], (X , V ) = Id V w (X , V ) Define for i = 2, , n y i = ( (X , V ) − 1)(1 − 2f i ) Let y = (y 1 , , y n ) and z = (0, y 2 , , y n ) Consider mappings P 1 , P 2 defined by…”

“…Classical references here are [6,7]. For recent results, we refer the reader to [1,2,17,18,20,32,35].…”

“…Also an application to minimal extensions with respect to the numerical radius will be presented. It is worth noticing that [1] gives a characterization of minimal numerical-radius extensions of operators from a normed linear space X onto its finite dimensional subspaces and comparison with minimal operator-norm extension.…”