Semigroups Generated by Similarity Orbits of Semi-invertible Operators and Unitary Orbits of Isometries

It is proved that for a left invertible operator A, which is not a sum of a scalar and a compact operator, the semigroup generated by similarity orbit of A contains all left invertible operators with suitable (large enough) deficiency. A corresponding result holds for right invertible operators. For isometries, the same statement is proved for unitary orbits.