Generalized Daugavet equations, affine operators and unique best approximation
Volume 238 / 2017
Studia Mathematica 238 (2017), 235-247
MSC: Primary 47L25, 47A50, 41A52; Secondary 46B20, 41A35.
DOI: 10.4064/sm8635-12-2016
Published online: 10 April 2017
Abstract
We introduce and investigate the notion of generalized Daugavet equation $\| A_1+\cdots +A_n\| =\| A_1\| +\cdots +\| A_n\| $ for affine operators $A_1,\ldots ,A_n$ on a reflexive Banach space into another Banach space. This extends the well-known Daugavet equation $\| T+I\| =\| T\| +1$, where $I$ denotes the identity operator. A new characterization of the Daugavet equation in terms of extreme points is given. We also present a result concerning uniqueness of best approximation.
