Pseudo regular elements in a normed ring
Richard F. Arens · Pacific Journal of Mathematics · 1992
Let A be an algebra, and let / be a linear mapping of A into some normed linear space C. For a in A we will write af for the image of a under /.By abf we mean (ab)f.Suppose \\abf\\ < M\\af\\ \\bf\\ for some real M, and all a, b in A .Then we will say that / is pseudo regular for A .We study mainly the case when C = A and A is a commutative Banach algebra.We present some conditions which imply pseudo regularity, and some that prevent it.For example, if the non-zero elements of the spectrum of / are bounded away from zero, then / is pseudo regular.A result (5.3) in the other direction is that if Σ!°oo I*/(OI<# < °o for a pseudo regular element / of L ι (Z), then the spectrum is bounded away from 0. Concerning the algebra C ι [a, b], any / which has no zero in common with its derivative is pseudo regular.