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Compactness Conditions for the Green Operator in L p (ℝ) Corresponding to a General Sturm–Liouville Operator
Author(s) -
Chernyavskaya N.,
Shuster L.
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200007)215:1<33::aid-mana33>3.0.co;2-6
Subject(s) - mathematics , operator (biology) , compact space , spectrum (functional analysis) , sturm–liouville theory , function (biology) , mathematical analysis , pure mathematics , physics , boundary value problem , quantum mechanics , chemistry , biochemistry , repressor , transcription factor , gene , evolutionary biology , biology
We consider the equation $-(r(x)y^{\prime} (x))^{\prime} + q(x)y(x) = f(x)\, , \quad x \, \in \, $ ℝ, where $r(x) > 0$ , $q(x) \ge 0$ for $x\, \in \,$ ℝ, $\textstyle {{1} \over {r(x)}} \in \, L^{\rm loc} _1$ (ℝ), $q(x) \, \in \, L^{\rm loc}_1$ (ℝ), $f(x) \, \in \, L_p$ (ℝ), $p \, \in \, [1, \, \infty]$ $(L_\infty$ (ℝ) := C (ℝ)). We give necessary and sufficient conditions under which, regardless of $p \, \in \, [1, \, \infty]$ , the following statements hold simultaneously: I) For any $f(x) \, \in \, L_p$ (ℝ) Equation (0.1) has a unique solution $y(x) \, \in \, L_p$ (ℝ) where $y(x)=(Gf)(x) \, $ $\mathop = ^{\rm def}\, $ \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\int ^{\infty}_{-\infty}$\end{document} $G(x,\, t)f(t)dt\, , \quad x \, \in \,$ ℝ. II) The operator $G: L_p$ (ℝ) → $L_p$ (ℝ) is compact. Here $G(x,\, t)$ is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.