Exponential dichotomy and $(L^p,L^q)$-admissibility

TL;DR

The paper characterizes exponential dichotomy for evolutionary families on the line with respect to a family of norms by the $(L^p,L^q)$-admissibility of the pair $(Y_1,Y_2)$, where $Y_1=L^p\cap C_b$ and $Y_2=L^q$ for $p \ge q$ and $(p,q) \neq (\infty,1)$. The central tool is the operator $H$ associated to the integral equation $x(t)=T(t,\tau)x(\tau)+\int_{\tau}^{t}T(t,s)y(s)\,ds$, whose bijectivity yields the admissibility-dichotomy equivalence and yields a unique stable/unstable projection pair. The results extend known characterizations to a broader $(p,q)$-range on the line, establish uniqueness of the dichotomic projections, and demonstrate robustness under small norm perturbations by a perturbation analysis that preserves the exponential dichotomy with the same norms. These findings broaden the applicability of admissibility-based dichotomy analysis in Banach-space evolution problems and provide a framework for stability under norm variations.

Abstract

We consider the notion of an exponential dichotomy with respect to a family of norms for an evolutionary family in a Banach space, and we characterize it by the admissibility of the pair $(L^p,L^q)$ for $p,q \in [1,\infty]$ with $p\ge q$. We then use this characterization to establish the robustness of an exponentially dichotomic evolutionary family with respect to a family of norms.

Theorems & Definitions (23)

• Definition 2.1
• Definition 2.2
• Remark 2.1
• Lemma 3.1
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Theorem 3.1
• proof