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

Abstract

For tuples of compact operators $\mathcal{T}=(T_1,\dots ,T_d)$ and $\mathcal{S}=(S_1,$ $\dots ,S_d)$ 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 ${\mathcal{F}}^d\mathcal{S}:=\{\text{z}\mathcal{S}:\text{z}\in {\mathbb{F}}^d\}.$ 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, $dist(\mathcal{T},{\mathbb{F}}^d\mathcal{S}{)}^p=\sum _{1\le i\le d}dist(T_i,\mathbb{F}{S}_i{)}^p.$ As a consequence, we deduce the equivalence of Birkhoff-James orthogonality, $\mathcal{T}{\perp }_B{\mathbb{F}}^d\mathcal{S}\iff T_i{\perp }_B{S}_i,$ under a sufficient condition. Furthermore, we explore the relation of one sided Gateaux 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.