Spectral and pseudospectral functions of various dimensions for symmetric systems
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- 作者：Vadim Mogilevskii
- 关键词：Symmetric differential system ; pseudospectral function ; Fourier transform ; m ; function ; dimension of a spectral function
- 刊名：Journal of Mathematical Sciences
- 出版年：2017
- 出版时间：March 2017
- 年：2017
- 卷：221
- 期：5
- 页码：679-711
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics, general;
- 出版者：Springer US
- ISSN：1573-8795
- 卷排序：221
文摘
The main object of the paper is a symmetric system Jy′ − B(t)y = ⋋∆(t)y defined on an interval Ι = [a, b) with the regular endpoint a. Let φ(⋅, λ) be a matrix solution φ(⋅, λ) of this system of an arbitrary dimension, and let \( \left( V\kern0.5em f\right)(s)={\displaystyle \underset{I}{\int }{\varphi}^{\ast}\left( t, s\right)\varDelta (t) f(t) d t} \) be the Fourier transform of the function f(⋅) ∈ LΔ2(I). We 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 \( {L}_{\varDelta}^2(I)\kern0.5em \mathrm{t}\mathrm{o}\kern0.5em {L}^2\left(\sigma; \kern0.5em {\mathbb{C}}^{n_{\sigma}}\right) \) with minimally possible kernel. Moreover, we find the minimally possible value of nσ and parametrize 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; Sakhnovich, Sakhnovich and Roitberg; Langer and Textorius.
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