Sums and products of cyclic operators

Abstract

It is proved that every bounded linear operator on a complex separable Hilbert space is the sum of two cyclic operators. For the product, we show that an operator T is the product of finitely many cyclic operators if and only if the kernel of T ∗ {T^ \ast } is finite-dimensional. More precisely, if dim ker T ∗ ≤ k ( 2 ≤ k > ∞ ) {T^ \ast } \leq k(2 \leq k > \infty ) , then T is the product of at most k + 2 k + 2 cyclic operators. We conjecture that in this case at most k cyclic operators would suffice and verify this for some special classes of operators.