Convolution Equations in Spaces of Polynomials on Locally Convex Spaces

Let E be a real or complex nuclear complete locally convex space and let P ( E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072577/94f683e7-9f6e-41e2-9c5e-98432cdb82df/content/eq127.tif"/> denote the space of the continuous polynomials on E, defined in a natural way as the direct sum of the spaces P ( n E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072577/94f683e7-9f6e-41e2-9c5e-98432cdb82df/content/eq128.tif"/> of the continuous homogeneous polynomials of degree n on ε, We prove that every non zero convolution operator on P ( E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072577/94f683e7-9f6e-41e2-9c5e-98432cdb82df/content/eq129.tif"/> is surjective.