The invariance of essential spectra of Balslev-Gamelin-Fashian differential operators in the scale of lebesgue spaces

Abstract

We consider spectral and semi-Fredholm properties of maximal and minimal Balslev-GamelinFashian differential operators in the scale of Lebesgue spaces on the half-line and, by way of application, obtain exact formulas for various essential spectra and the spectrum of ordinary differential operators with polynomial coefficients whose order does not exceed the order of the corresponding derivative. Balslev and Gamelin [1, p. 771] investigated Fredholm properties of maximal differential operators generated by the Fashian differential expression of the form (mf)(t) = ~j~=o aj(t)f(3)(t), where ak(t) = O (tk), in the spaces LV(1, oc), 1 < p < r Various essential spectra of the maximal and minimal Euler differential operators generated by the expression m with coefficients ak(t) = akt k in the spaces LP(1, co) and LP(0, 1), 1 < p < co, were given for the first time in [2].