DIFFERENTIAL OPERATORS OF INFINITE ORDER IN THE SPACE OF RAPIDLY DECREASING SEQUENCES

doi:10.1016/S0252-9602(16)30002-9

Abstract

Abstract:

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)=φ_kD^k.φ_k constant numbers an a power of D. Dⁿ, meaning, is there a isomorphism X (from s onto s) such that X_φ(D)=DⁿX?. 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.

Key words: Ordinary differential operators, sequence spaces, operators on function spaces

CLC Number:

47E05

M. MALDONADO, J. PRADA, M. J. SENOSIAIN. DIFFERENTIAL OPERATORS OF INFINITE ORDER IN THE SPACE OF RAPIDLY DECREASING SEQUENCES[J].Acta mathematica scientia,Series B, 2016, 36(2): 325-333.

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