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A set $\mathcal{S}$ of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists $A\in \mathcal{S}$ such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are: (1) Every irreducible semitransitive Jordan algebra is actually transitive. (2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]