Function Theory and necessary conditions for a Schwarz lemma related to μ-Synthesis Domains

2025/10/28 by Keshari, Dinesh Kumar, Mandal, Shubhankar, Pal, Avijit · 3 citations
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2510.24555

Abstract

A subset of ℂ⁷ (respectively, of ℂ⁵) associated with the structured singular value μ_E, defined on 3 × 3 matrices, is denoted by G_E(3;3;1,1,1) (respectively, by G_E(3;2;1,2)). In control engineering, the structured singular value μ_E plays a crucial role in analyzing the robustness and performance of linear feedback systems. We characterize the domain G_E(3;3;1,1,1) and its closure Γ_E(3;3;1,1,1), and employ realization formulas to describe both. The domain G_E(3;3;1,1,1) and its closure are neither circular nor convex; however, they are simply connected. We provide an alternative proof of the polynomial and linear convexity of Γ_E(3;3;1,1,1). Furthermore, we establish necessary conditions for a Schwarz lemma on the domains G_E(3;3;1,1,1) and G_E(3;2;1,2), and describe the relationships between these two domains as well as between their closed boundaries.

Cited by

• Functional Models for Γ_{E(3; 3; 1, 1, 1)}-contraction, Γ_{E(3; 2; 1, 2)}-contraction and Tetrablock contraction
• Necessary Conditions for Γ_{E(3; 3; 1, 1, 1)}-Isometric Dilation, Γ_{E(3; 2; 1, 2)}-Isometric Dilation and \mathcalP-Isometric Dilation
• Canonical Decompositions and Conditional Dilations of Γ_{E(3; 3; 1, 1, 1)}-Contraction and Γ_{E(3; 2; 1, 2)}-Contraction

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