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* is finite-dimensional. More precisely, if dimker T* less than or equal to k (2 less than or equal to k < infinity), then T is the product of at most 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.