A note on generalized projections in $c_{0}$

Beata Deręgowska¹ ; Barbara Lewandowska¹

¹ Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-048 Kraków, Poland

Annales Polonici Mathematici, Tome 111 (2014) no. 1, pp. 59-72

Résumé

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.

Détail

DOI : 10.4064/ap111-1-5

Keywords: subset subspaces banach space define set generalized projections mathcal mathcal infty mathop ker * cap infty main paper discuss existence uniqueness strong uniqueness minimal generalized projection formulas relative generalized projection constant strong uniqueness constant given blatter cheney ann mat pura appl lewicki micek approx theory where projections has considered discuss real complex cases