The operatorshnandinand their dualshmacr;nandimacr;ndefined on the free semigroupX+for a nonempty setXby Gerhard and Petrich are proved here to be homomorphisms of onto certain subsets ofX+with a suitable multiplication. From the work of the authors mentioned, these operators induce fully invariant congruences onX+corresponding to join irreducible varieties of bands ifXis countably infinite. New operators onX+are defined by means of these operators which give homomorphisms in an analogous way and induce fully invariant congruences onX+corresponding to all varieties of bands except for the variety of all bands, and some varieties of normal bands. The former of these was investigated by the mentioned authors and the latter must be treated differently. By means of the above operators we are able to characterize all cases of relatively free bands.