Spectral and pseudospectral functions of various dimensions for symmetric systems

Abstract

The main object of the paper is a symmetric system Jy ′ − B(t)y = λ∆(t)y defined on an interval I = [a, b) with the regular endpoint a. Let ϕ(·, λ) be a matrix solution of this system of an arbitrary dimension and letWe define a pseudospectral function of the system as a matrix-valued distribution function σ(·) of the dimension nσ such that V is a partial isometry from L 2 ∆ (I) to L 2 (σ; C nσ ) with the minimally possible kernel. Moreover, we find the minimally possible value of nσ and parameterize all spectral and pseudospectral functions of every possible dimensions nσ by means of a Nevanlinna boundary parameter. The obtained results develop the results by Arov and Dym; A. Sakhnovich, L. Sakhnovich and Roitberg; Langer and Textorius.The system is called regular if b < ∞ and I ||B(t)|| dt < ∞, I ||∆(t)|| dt < ∞ .

On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight

“…Below we suppose that U ∈ B(C n , C p ) is an operator satisfying (3.6). Then the following assertion holds (see [27,Lemma 3.3]).…”

On uniform convergence of the inverse Fourier transform for differential equations and Hamiltonian systems with degenerating weight

“…Below we suppose that $$U\in \mbox{\boldmath $B$}\big (\mathbb {C}^n,\mathbb {C}^p\big )$$ is an operator satisfying (3.6). Then the following assertion holds (see [30, Lemma 3.3]). Assertion The equality $$\begin{gather} T=\big \lbrace \widetilde{\pi }_\Delta \lbrace y, f\rbrace : \lbrace y,f\rbrace \in \mathcal {T}_{\max },\; Uy(a)=0 \;\;{\rm and}\; \;[y,z]_b=0,\; z\in {\rm dom}\,\mathcal {T}_{\max }\big \rbrace \end{gather}$$defines a (closed) symmetric extension T of $$T_{\min }$$.…”