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

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 $1\leq i\leq d.$ We prove that if $p=\infty,$ then $dist(\mathcal{T},\mathbb{F}^d\mathcal{S})=\underset{1\leq i\leq d}{\max}dist(T_i,\mathbb{F}S_i),$ and for $1\leq p<\infty,$ under a sufficient condition, $dist(\mathcal{T},\mathbb{F}^d\mathcal{S})^p=\underset{1\leq i\leq d}{\sum}dist(T_i,\mathbb{F}S_i)^p.$ 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 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.