We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D）...We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D） =^∞∑k=0φkD^k.φk constant numbers an a power of D.Dn, meaning, is there a isomorphism X （from s onto s） such that Xφ（D） = D^nX？. We prove that if φ（D） is equivalent to Dn, then φ（D） is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n = 1.展开更多
文摘We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D） =^∞∑k=0φkD^k.φk constant numbers an a power of D.Dn, meaning, is there a isomorphism X （from s onto s） such that Xφ（D） = D^nX？. We prove that if φ（D） is equivalent to Dn, then φ（D） is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n = 1.
