Minimal Boolean sum and blending-type projections and extensions (Q1570062)

Let \(V\) and \(W\) be finite-dimensional subspaces of a Banach space \(X\). Let \(A:V\to [v_1,\dots, v_n]= V\), \(B:W\to[w_1,\dots, w_m]= W\) be finite operators on \(V\) and \(W\) respectively, and let \(P:X\to V\), \(Q: X\to W\) denote two bounded extension operators \(A\) and \(B\), respectively. Consider the Boolean sum of \(P\) and \(Q\) \[ P\oplus Q= P+Q- PQ:X\to V+ W=Z. \] Suppose that \(Q= Q_0\) is fixed and let \({\mathcal R}= \{P\oplus Q_0\}\). In the present paper the authors characterize the solution of the following extremal problem (minimal Boolean sum): \[ \min_{R\in{\mathcal R}}\|R\|= \min_P\|P\oplus Q_0\|_{X\to Z}, \] and a similar problem for blending-type projections.