Dissipative ordinary differential operators of even order

Dissipative Ordinary Differential Operators of Even Order

“…Moreover, these (lb -km) + (la -km) functions span the space of all solutions which are square integrable at both ends. Also the space of solutions of (4.4) which are square integrable at both ends, i.e., which belong to L2(a, b; F) is of dimension (ma -km) + (mb -km), and again it can be shown as in §1.3 of [5] that the square integrable solutions of (4.3) and (4.4) comprise all of the solutions of equation (F-^-IXF^Llx + ^y1 = 0.…”

“…In [5], Phillips and the present writer have considered the first-and second-order cases by a general operator-theoretic approach developed by Phillips in [2] and [3]. That it has been possible, in the general case treated in the present paper, to obtain all the information obtained in [1], [4], [5] by using the operator-theoretic approach is an indication of the power of that method. The author wishes to express his gratitude to Professor Ralph Phillips for suggesting this more general investigation, for his inspiration, and for his helpful criticisms and encouragement in the preparation of this paper, and also to the referee for his suggestions.…”

“…This concludes the proof. This corollary can be proved in a way analogous to the method used in proving the corollary to Lemma 1.2.1 of [5].…”

“…In order to determine the set of solutions of (4.2), we consider the two equations Arguing now as in §1.3 of [5], we construct a special basis for the set of all square integrable solutions of (4.3). Let Na and Nb he a fixed ¿m-dimensional and a fixed ¿«z-dimensional subspace contained in Ca and Cb, respectively.…”