Pseudo regular elements in a normed ring

Richard Arens

Vol. 154 (1992), No. 1, 1–15

DOI: 10.2140/pjm.1992.154.1

Abstract

Let A be an algebra, and let f 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 f. 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 f 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 f are bounded away from zero, then f is pseudo regular. A result (5.3) in the other direction is that if ∑ _−∞^∞|tf(t)|dt < ∞ for a pseudo regular element f of L¹(ℤ), then the spectrum is bounded away from 0. Concerning the algebra C¹[a,b], any f which has no zero in common with its derivative is pseudo regular.

Mathematical Subject Classification 2000

Primary: 46J05

Milestones

Received: 4 March 1991

Revised: 15 May 1991

Published: 1 May 1992