Finite Rank Operators in Certain Algebras

http://dx.doi.org/10.4153/CMB-1999-053-3
Canad. Math. Bull. 42(1999), 452-462

  Published:1999-12-01
 Printed: Dec 1999

Let $\Alg(\l)$ be the algebra of all bounded linear operators on a normed linear space $\x$ leaving invariant each member of the complete lattice of closed subspaces $\l$. We discuss when the subalgebra of finite rank operators in $\Alg(\l)$ is non-zero, and give an example which shows this subalgebra may be zero even for finite lattices. We then give a necessary and sufficient lattice condition for decomposing a finite rank operator $F$ into a sum of a rank one operator and an operator whose range is smaller than that of $F$, each of which lies in $\Alg(\l)$. This unifies results of Erdos, Longstaff, Lambrou, and Spanoudakis. Finally, we use the existence of finite rank operators in certain algebras to characterize the spectra of Riesz operators (generalizing results of Ringrose and Clauss) and compute the Jacobson radical for closed algebras of Riesz operators and $\Alg(\l)$ for various types of lattices.