Min–max relations for tuples of operators in terms of component spaces

Abstract

For tuples of compact operators \(\mathcal {T}=(T_1,\ldots , T_d)\) and \(\mathcal {S}=(S_1,\ldots ,S_d)\) on Banach spaces over a field \(\mathbb {F}\), considering the joint p-operator norms on the tuples, we study \(dist(\mathcal {T},\mathbb {F}^d\mathcal {S}),\) the distance of \(\mathcal {T}\) from the d-dimensional subspace \(\mathcal {F}^d\mathcal {S}:=\{{\textbf {z}}\mathcal {S}:{\textbf {z}}\in \mathbb {F}^d\}.\) We obtain a relation between \(dist(\mathcal {T},\mathbb {F}^d\mathcal {S})\) and \(dist(T_i,\mathbb {F}S_i),\) for \(1\le i\le d.\) We prove that if \(p=\infty ,\) then \(dist(\mathcal {T},\mathbb {F}^d\mathcal {S})=\underset{1\le i\le d}{\max }dist(T_i,\mathbb {F}S_i),\) and for \(1\le p<\infty ,\) under a sufficient condition, \(dist(\mathcal {T},\mathbb {F}^d\mathcal {S})^p=\underset{1\le i\le d}{\sum }dist(T_i,\mathbb {F}S_i)^p.\) As a consequence, we deduce the equivalence of Birkhoff-James orthogonality, \(\mathcal {T}\perp _B \mathbb {F}^d\mathcal {S} \Leftrightarrow T_i\perp _B S_i,\) under a sufficient condition. Furthermore, we explore the relation of one sided Gâteaux derivatives of \(\mathcal {T}\) in the direction of \(\mathcal {S}\) with that of \(T_i\) in the direction of \(S_i.\) Applying this, we explore the relation between the smoothness of \(\mathcal {T}\) and \(T_i.\) By identifying an operator, whose range is \(\ell _\infty ^d,\) as a tuple of functionals, we effectively use the results obtained here for operators whose range is \(\ell _\infty ^d\) and deduce nice results involving functionals.