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

2025/03/13 by Mal, Arpita
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2503.10128

Abstract

For tuples of compact operators T=(T₁,…, T_d) and S=(S₁, …,S_d) on Banach spaces over a field \mathbbF, considering the joint p-operator norms on the tuples, we study dist(T,\mathbbF^dS), the distance of T from the d-dimensional subspace F^dS:=\zS:z∈ \mathbbF^d\. We obtain a relation between dist(T,\mathbbF^dS) and dist(T_i,\mathbbFS_i), for 1≤ i≤ d. We prove that if p=∞, then dist(T,\mathbbF^dS)=\underset1≤ i≤ dmaxdist(T_i,\mathbbFS_i), and for 1≤ p<∞, under a sufficient condition, dist(T,\mathbbF^dS)^p=\underset1≤ i≤ d∑dist(T_i,\mathbbFS_i)^p. As a consequence, we deduce the equivalence of Birkhoff-James orthogonality, T⊥_B \mathbbF^dS ⇔ T_i⊥_B S_i, under a sufficient condition. Furthermore, we explore the relation of one sided Gateaux derivatives of T in the direction of S with that of T_i in the direction of S_i. Applying this, we explore the relation between the smoothness of T and T_i. By identifying an operator, whose range is ℓ_∞^d, as a tuple of functionals, we effectively use the results obtained here for operators whose range is ℓ_∞^d and deduce nice results involving functionals.

Related

• Subdifferential of the B(H,K) norm, and approximate orthogonality
• Birkhoff--James orthogonality of operators in semi-Hilbertian spaces and\n its applications
• On Best approximations to compact operators
• A distance formula for tuples of operators
• An approximation problem in the space of bounded operators