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Minimal projections onto hyperplanes in vector-valued sequence spaces. (English) Zbl 1328.47033

Let \(V\) be a subspace of a Banach space \(X\) and \(L(X,V)\) the space of all linear continuous mappings \(P: X\to V\). Consider the projections of \(X\) on \(V\) (i.e., the maps \(P\in L(X,V)\) such that \(P/V = \operatorname{id}\)) and the minimal projections (projections of minimal norm). The authors study projections onto hyperplanes \(V\) in vector-valued sequence spaces \(X\) equipped by the \(c_0\) or \(\ell_1\) norm. They give an estimate of the norm of \(P\), a lower bound for the projection constant and, in the case of \(c_0\) norm, characterizations of one-complemented hyperplanes. They also present some consequences of the obtained results.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46E40 Spaces of vector- and operator-valued functions
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47A58 Linear operator approximation theory
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References:

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