Sums of strongly irreducible operators (Q2705804)

A Hilbert space operator \(T\) is strongly irreducible if every operator similar to \(T\) has no non-trivial reducing subspace. It is proved that every operator on an infinite dimensional space is the sum of three strongly irreducible operators. This is done by writing the operator as the sum of three operators each of which is a multiple of an operator close to the simple unilateral shift or its adjoint. Then it is shown that operators in certain special classes (e.g., compact operators, multicyclic operators, bilateral weighted shifts, triangular operators, analytic Toeplitz operators) can be written as the sum of two strongly irreducible operators.