Some Results on Absolute Projection Constants

Abstract

Let V be an n-dimensional real or complex Banach space and let $\lambda (V)$ denote its absolute projection constant. For any $N\in \mathbb{N},$ $N\ge n$ define $${\lambda }_n^N=\sup \{\lambda (V):dim(V)=n,V\subset l_{\infty }^{\left(N\right)}\}.$$ In this article, we determine, in the complex case, minimal projections with respect to l₁-norm as well as with respect to $l_{\infty }$-norm for subspaces given by solutions of certain extremal problems. As an application we show that for any $n,N\in \mathbb{N},$ $N\ge n$, there exists an n-dimensional subspace $V_n\subset l_1^{(N)}$ such that $${\lambda }_n^N=\lambda (V_n,l_1^{\left(N\right)}).$$ This generalizes the results from Castejon and Lewicki (2014) obtained in the real case.

Keywords:

• Absolute projection constant
• minimal projection
• symmetric matrices
• Hermitian matrices
• unitary matrices
• eigenvalues
• eigenvectors

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 46A22
• 47A30
• 47A58