Some uniqueness theorems for differential equations with operator coefficients
Robert Carroll, J. Neuwirth · Transactions of the American Mathematical Society · 1964
where u(i) du then Ai determines an operator in Y(H) by extension to H which we denote again by Ai. We will regard A as the basic operator and restrict the behavior of the Ai relative to A; Am is presumed to be densely defined and we note that A-' c E (H) implies A is closed since un -+ u and Aun -+ v imply u = -v. It is seen also that Ak, k < m, is now densely defined and closed. In practice the basic operator will be Am and if Am = A is closed with j .AR(AZ A) <M < oo then a closed A'I' can be defined (see [5]). Suppose now only that u(i) D(Aml-) and set ui+1 = Am i ) (i = 0,... ,m 1). Let Ai be bounded operators (which may or may not arise from the preceding considerations) and assume Ai A c AA, with all ii and E commuting. (These hypotheses will hold throughout the paper along with A -1 E Y(H) and D(Am) dense.) Then (1.1) leads us to pose the following problem for t -+ u(t) e'(H), t -+ Am-() E eo(H)