Gordon MacDonald

An irreducible semigroup of non-negative square-zero operators

We construct an irreducible multiplicative semigroup of non-negative square-zero operators acting onLp[0,1), for 1≤p<∞.

On transitive linear semigroups

Abstract This paper deals with semigroups of linear transformations which act transitively on a finite-dimensional vector space. An explicit canonical form is obtained for the semigroups which lack proper transitive left ideals. The class of such semigroups can be considered to be an extention of the class of transitive groups. It contains all minimal transitive (and hence all sharply transitive) semigroups.

A spatial version of Wedderburn’s Principal Theorem

In this article we verify that ‘Wedderburn’s Principal Theorem’ has a particularly pleasant spatial implementation in the case of cleft subalgebras of the algebra of all linear transformations on a finite-dimensional vector space. Once such a subalgebra is represented by block upper triangular matrices with respect to a maximal chain of its invariant subspaces, after an application of a block upper triangular similarity, the resulting algebra is a linear direct sum of an algebra of block-diagonal matrices and an algebra of strictly block upper triangular matrices (i.e. the radical), while the block-diagonal matrices involved have a very nice structure. We apply this result to demonstrate that, when the underlying field is algebraically closed, and , the algebra is unicellular, i.e. the lattice of all invariant subspaces of is totally ordered by inclusion. The quantity stands for the length of (every) maximal chain of non-zero invariant subspaces of .

MULTIPLICATIVE DIAGONALS OF MATRIX SEMIGROUPS

We consider semigroups of matrices where either the diagonal map or the diagonal product map is multiplicative, and deduce structural properties of such semigroups.

Common fixed points and common eigenvectors for sets of matrices

The following questions are studied: Under what conditions does the existence of a (nonzero) fixed point for every member of a semigroup of matrices imply a common fixed point for the entire semigroup? What is the smallest number k such that the existence of a common fixed point for every k members of a semigroup implies the same for the semigroup? If every member has a fixed space of dimension at least k: What is the best that can be said about the common fixed space? We also consider analogs of these questions with general eigenspaces replacing fixed spaces.

Operator Semigroups for which Reducibility Implies Decomposability

AbstractA description of the lattice of invariant subspaces is provided for multiplicative semigroups

Cone-transitive matrix semigroups

\mathcal{S}

Principal-ideal Bands

of bounded operators on Lp(X,μ) which are closed under multiplication on the left or right by bounded multiplication operators. Applications are then given to semigroups of positive quasinilpotent operators.