Minimal extensions in direct sums of Banach spaces

Let P(X, Y ) denote the set of all continuous, linear projections from X onto its subspace Y. Put λ(Y,X) = inf{∥P∥ : P ∈ P(X, Y )}. A projection Po ∈ P(X, Y ) is called a minimal projection if and only if ∥P∥ = λ(Y,X). The aim of this paper is to give some estimates of λ(Y,X) in the case when X = Mn j=1 Xj , where (Xj , ∥·∥j) for j = 1, ..., n, are Banach spaces and Y ⊂ X is a closed subspace of X equipped with a norm ∥x∥ = g(∥x1∥1, ..., ∥xn∥n), where g is a monotone norm on Kn, K = C or K = C. We present our results in a more general case of minimal extensions (see Definition (2)). This approach leads to some new estimates of norms of minimal extensions. Also the problems of unicity and strong unicity of minimal extensions (see Definition(3)) is studied.