Isomorphism theorems for infinite order differential operators

Author:
J. D. Buckholtz

Journal:
Proc. Amer. Math. Soc. **40** (1973), 533-538

MSC:
Primary 47E05; Secondary 34A35

DOI:
https://doi.org/10.1090/S0002-9939-1973-0320820-5

MathSciNet review:
0320820

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Abstract: The operators studied are infinite order linear differential operators with constant coefficients. We require that the characteristic function of the operator be analytic in a neighborhood of 0 and have a radius of convergence which is less than that of its reciprocal. It is shown that such operators may be regarded as isomorphisms between Banach spaces of entire functions. These Banach spaces, in every case, are isomorphic to the sequence space . Further, translation in the domain plane defines an automorphism on both the domain and range space of the operator.

**[1]**J. D. Buckholtz,*Appell polynomial expansions and biorthogonal expansions in Banach spaces*, Trans. Amer. Math. Soc. (to appear). MR**0333210 (48:11535)****[2]**-,*Appell polynomials whose generating function is meromorphic on its circle of convergence*, Bull. Amer. Math. Soc.**79**(1973), 469-472. MR**0315128 (47:3677)****[3]**-,*Appell polynomials and differential equations of infinite order*, Trans. Amer. Math. Soc. (to appear). MR**0352456 (50:4943)****[4]**R. D. Carmichael,*On nonhomogeneous linear differential equations of infinite order with constant coefficients*, Amer. J. Math.**58**(1936), 473-486. MR**1507170****[5]**I. M. Sheffer,*Systems of linear differential equations of infinite order, with constant coefficients*, Ann. of Math.**30**(1928/29), 250-264. MR**1502879**

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0320820-5

Article copyright:
© Copyright 1973
American Mathematical Society