TY - JOUR
T1 - Compactness conditions for the green operator in Lp(double-struck R sign) corresponding to a general Sturm-Liouville operator
AU - Chernyavskaya, N.
AU - Shuster, L.
PY - 2000/1/1
Y1 - 2000/1/1
N2 - We consider the equation (0.1) -(r(x)y′(x))′ + q(x)y(x) = f(x), x ∈ double-struck R sign, where r(x) > 0, q(x) ≥ 0 for x ∈ double-struck R sign, 1/r(x) ∈ Lloc1(double-struck R sign), q(x) ∈ Lloc1 (double-struck R sign), f(x) ∈ Lp(double-struck R sign), p ∈ [1, ∞] (L∞(double-struck R sign) := C(double-struck R sign)). We give necessary and sufficient conditions under which, regardless of p ∈ [1, ∞], the following statements hold simultaneously: I) For any f(x) ∈ Lp(double-struck R sign) Equation (0.1) has a unique solution y(x) ∈ Lp(double-struck R sign) where y(x) = (Gf)(x) def= ∫∞-∞ G(x,t)f(t)dt, x ∈ double-struck R sign. II) The operator G : Lp (double-struck R sign) → Lp(double-struck R sign) 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.
AB - We consider the equation (0.1) -(r(x)y′(x))′ + q(x)y(x) = f(x), x ∈ double-struck R sign, where r(x) > 0, q(x) ≥ 0 for x ∈ double-struck R sign, 1/r(x) ∈ Lloc1(double-struck R sign), q(x) ∈ Lloc1 (double-struck R sign), f(x) ∈ Lp(double-struck R sign), p ∈ [1, ∞] (L∞(double-struck R sign) := C(double-struck R sign)). We give necessary and sufficient conditions under which, regardless of p ∈ [1, ∞], the following statements hold simultaneously: I) For any f(x) ∈ Lp(double-struck R sign) Equation (0.1) has a unique solution y(x) ∈ Lp(double-struck R sign) where y(x) = (Gf)(x) def= ∫∞-∞ G(x,t)f(t)dt, x ∈ double-struck R sign. II) The operator G : Lp (double-struck R sign) → Lp(double-struck R sign) 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.
UR - http://www.scopus.com/inward/record.url?scp=0040184101&partnerID=8YFLogxK
U2 - 10.1002/1522-2616(200007)215:1<33::aid-mana33>3.0.co;2-6
DO - 10.1002/1522-2616(200007)215:1<33::aid-mana33>3.0.co;2-6
M3 - Article
AN - SCOPUS:0040184101
SN - 0025-584X
VL - 215
SP - 33
EP - 53
JO - Mathematische Nachrichten
JF - Mathematische Nachrichten
ER -