A spectral notion for dynamic equations on time scales
The author considers the linear \(N\)-dimensional system of dynamic equations \[ x^\Delta= A(t) x\tag{1} \] on a time scale \(\mathbb{T}\) with an \(rd\)-continuous and regressive matrix-function \(A: \mathbb{T}\to \mathbb{R}^{N\times N}\), \(N\in\mathbb{N}= \{1,2,\ldots\}\). It is assumed that \(\mathbb{T}\) is unbounded to the right and either bounded to the left (one sided-time) or unbounded to the left (two-sided time), and \(\phi_A(t,s)\) is the evolution operator of (1). The notion of invariant vector bundle of (1) is defined to be a nonempty set \(W\subset\mathbb{T}\times \mathbb{R}^N\) with the properties: (a) for every \(\tau\in\mathbb{T}\) the fiber \(W(\tau)= \{\xi\in\mathbb{R}^N: (\tau,\xi)\in W\}\) is a linear subspace of \(\mathbb{R}^N\), (b) it is invariant, i.e., \((\tau,\xi)\in W\Rightarrow (t,\phi_A(t,\tau)\xi)\in W\) for all \(t\in\mathbb{T}\). An invariant projector of (1) is also defined to be a function \(P:\mathbb{T}\to \mathbb{R}^{N\times N}\) of projection \(P(t)\), \(t\in\mathbb{T}\), such that \[ P(t)\phi_A(t, s)= \phi_A(t,s) P(s)\quad\text{for }t,s\in\mathbb{T}. \] For \(\gamma\in \mathbb{R}\), it is said that (1) admits an exponential dichotomy with growth rate \(\gamma\) (\(\gamma\)-ED) if there exist an invariant projector \(P:\mathbb{T}\to \mathbb{R}^{N\times N}\) and constants \(K\geq 1\) and \(\alpha> 0\) such that, for \(t,s\in\mathbb{T}\), the dichotomy estimates \[ \|\phi_A(t,s)P(s)\|\leq ke^{(\gamma-\alpha)(t-s)}\quad\text{for }t\geq s\qquad\text{and} \] \[ \|\phi_A(t, s)(I- P(s)\|\leq ke^{(\gamma+\alpha)(t-s)}\quad\text{for }t\leq s, \] hold. The dichotomy spectrum of (1) is defined to be the set \[ \Sigma(A)= \{\gamma\in \mathbb{R}: x^\Delta= A(t) x\text{ admits no }\gamma\text{-ED}\} \] and the resolvent set \(\rho(A)= \mathbb{R}\setminus \Sigma(A)\) is its complement. The main result is the spectral theorem, which states that the dichotomy spectrum \(\Sigma(A)\) of (1) is the disjoint union of \(n\) closed intervals (called spectral intervals) where \(0\leq n\leq N\), i.e., \(\Sigma(A)=\emptyset\) or \(\Sigma(A)= \mathbb{R}\) or one of the four cases \[ \Sigma(A)= \left\{\begin{matrix} [a_1,b_1]\\ \text{or}\\ (-\infty,b_1]\end{matrix}\right\}\cup [a_2,b_2]\cup\cdots\cup [a_{n-1}, b_{n-1}]\cup \left\{\begin{matrix} [a_n, b_n]\\ \text{or}\\ [a_n,\infty)\end{matrix}\right\}, \] where \(a_1\leq b_1< a_2\leq b_2<\cdots< a_n\leq b_n\). In the autonomous case with \(\mathbb{T}= \mathbb{R}\) these intervals reduce to the real parts of the eigenvalues of \(A\). In any case, the spectral intervals are associated with invariant vector bundles comprising solutions with a common exponential growth rate.
scientific article; zbMATH DE number 1758484