Semigroups generated by similarity orbits of semi-invertible operators and unitary orbits of isometries (Q1434125)

Let \(H\) be a separable infinite-dimensional Hilbert space and \(A\) a left invertible operator with deficiency \(d\equiv\dim\ker A^*\), which is not a sum of a scalar and a compact operator. It is proved that every left invertible operator with deficiency \(48d\) is a product of 48 operators similar to \(A\). A corresponding result holds for right invertible operators. Similar results are proved for isometries and unitary operators.