Distance formulae and best approximation in the space of compact operators
This paper is divided into four sections. The introduction describes the main focus of the paper and motivates the distance formulae derived. The authors also review several useful concepts, such as the definitions of strictly convex and smooth spaces, Birkhoff-James orthogonality, and semi-inner product spaces. The introduction ends with two theorems. The first theorem gives a characterization of best approximants of a point in a Banach space from a finite dimensional subspace, and the second gives a description of the extreme points of the dual of the space of compact operators between two Banach spaces. Section 2 contains theorems giving the distance of an element in a Banach space from a finite dimensional subspace. One of them gives a formula for the distsnce of a compact operator \(T\) from an $n$-dimensional subspace of compact operators. The section ends with an example that computes the set of best approximants of a rank~2 projection \(T\) on \(c_0\) and the span of two rank~3 operators on \(c_0\). Section 3 presents distance formulae in the setting of compact operators \(\mathbb{K} (X,Y)\) in terms of a compatible semi-inner product on \(Y\). For an example of the type of results presented we quote Theorem~3.4. {Theorem.} Let \(X\) be a reflexive strictly convex Banach space and \(Y\) an arbitrary Banach space. Let \(T\in \mathbb{K} (X,Y)\) and \(\mathcal{W}\) a subspace of \(\mathbb{K} (X,Y)\) such that \(\dim (\mathcal{W})=n\) and \(T \notin \mathcal{W}\). Then there exist a semi-inner product on \(Y\) and positive scalars \(\lambda_1, \dots, \lambda_h\) (with \(h\leq n+1\) for real spaces and \(h\leq 2n+1\) for complex spaces) such that \(\sum_{i=1}^h \lambda_i =1\) and \( \operatorname{dist} (T, \mathcal{W})=\max \sum_{i=1}^h \lambda_i [Tx_i, y_i],\) with \( x_i \in \operatorname{ext}(X),\) \( \|y_i\|=1\), \(\sum_{i=1}^h \lambda_i \Im([Tx_i, y_i])=0\), and \(\sum_{i=1}^h \lambda_i [Sx_i, y_i]=0\) for all \(S\in \mathcal{W}. \) The last section presents applications of the distance formulae to the study of approximate Birkhoff-James orthogonality.
(approximate) Birkhoff-James orthogonality
scientific article; zbMATH DE number 7461204
scientific article