Open AccessON THE ESSENTIAL SPECTRA OF GENERAL DIFFERENTIAL OPERATORS
Author(s) -
Sobhy El-Sayed Ibrahim
Publication year - 1999
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.30.1999.4213
Subject(s) - mathematics , lambda , interval (graph theory) , differential operator , spectrum (functional analysis) , order (exchange) , operator (biology) , ordinary differential equation , essential spectrum , limit (mathematics) , pure mathematics , hilbert space , mathematical analysis , combinatorics , discrete mathematics , differential equation , quantum mechanics , physics , biochemistry , chemistry , finance , repressor , transcription factor , economics , gene
In this paper, it is shown in the cases of one and two singular end-points and when all solutions of the equation $M[u]-\lambda uw=0$, and its adjoint $M^+[v] -\lambda wv = 0$ are in $L_w^2 (a, b)$ (the limit circle case) with $f\in L^2_w(a,b)$ for $M[u]-\lambda wu=wf$ that all well-posed extensions of the minimal operator $T_0(M)$ generated by a general ordinary quasi-differential expression $M$ of $n$-th order with complex coefficients have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly slovable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression $M$ studied in [1] and [12], and also extend those proved in [8] in the case of one singular end-point of the interval [a,b).