A note on generalized projections in $c_{0}$
Volume 111 / 2014
Abstract
Let $V \subset Z$ be two subspaces of a Banach space $X$. We define the set of generalized projections by $$ \mathcal {P}_V(X,Z):=\{ P \in \mathcal {L}(X,Z): P|_V ={\rm id} \} . $$ Now let $X=c_0$ or $l^m_\infty ,$ $Z:=\mathop {\rm ker}f$ for some $f\in X^* $ and $V:=Z\cap l^{n}_\infty $ $(n < m).$ The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215–227] and G. Lewicki and A. Micek [J. Approx. Theory 162 (2010), 2278–2289] where the case of projections has been considered). We discuss both the real and complex cases.
