Decomposability of Finite-Rank Operators in Commutative Subspace Lattice Algebras

In this paper, we explore conditions for a finite-rank operator in a commutative subspace lattice (CSL) algebra to be decomposable (that is, it can be written as the sum of rank one operators in that algebra). We introduce correlation coefficients for rank two operators, the property (F), and correlation matrices for finite-rank operators, based on which we prove that a rank two operator is decomposable if and only if it has only finitely many correlation coefficients, and if a finite-rank operator has the property (F) (has only finitely many correlation matrices) then it is decomposable.