Showing papers on "Idempotence published in 1985"

Products of idempotent linear transformations

Abstract: In 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

The ideal structure of idempotent-generated transformation semigroups

Abstract: Let X be a set and the semigroup (under composition) of all total transformations from X into itself. In ([6], Theorem 3) Howie characterised those elements of that can be written as a product of idempotents in different from the identity. We gather from review articles that his work was later extended by Evseev and Podran [3, 4] (and independently for finite X by Sullivan [15]) to the semigroup of all partial transformations of X into itself. Howie's result was generalized in a different direction by Kim [8], and it has also been considered in both a topological and a totally ordered setting (see [11] and [14] for brief summaries of this latter work). In addition, Magill [10] investigated the corresponding idea for endomorphisms of a Boolean ring, while J. A. Erdos [2] resolved the analogous problem for linear transformations of a finite–dimensional vector space.

I. Idempotent Multipliers of H¹ on the Circle. II. A Mean Oscillation Inequality for Rearrangements

Abstract: H1(T) is the space of integrable functions f on the circle T such that the Fourier coefficients f(n) vanish for negative integers n. A multiplier is by definition a map m of H1 to itself such that the Fourier transform diagonal izes m. Let m(n) denote the diagonal coefficients of m for nonnegative n. Then m is called idempotent if each coefficient is zero or one. Theorem: If m is idempotent, then the set of n for which m(n) = 1 is a finite Boolean combination of sets of nonnegative integers of the following three types: finite sets, arithmetic sequences, and lacunary sequences. By definition, a sequence is lacunary if there is a real number q > 1 such that each term of the sequence is at least as large as q times the preceding term. The theorem implies a classification of the projections in H1 which commute with translations, or, what is equivalent on the circle (but not on the line), of the closed, translation invariant subspaces which are complemented in H1. In the course of the proof, a 1 ower bound is obtai ned on the operator norm of a multiplier whose coefficients are 0 or greater than 1 in magnitude. This bound implies that the number of nonzero coefficients in disjoint intervals of the same length is the same, up to some factor depending on the norm of m, provided that both intervals are shorter than their distance from 0. Part II is unrelated to Part I. There it is proved that a general expression measuring the oscillation of a function on an interval is minimized by the decreasing rearrangement of the function. A special case of this expression is the BMO norm for functions of bounded mean oscillation.