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Concerning the Cesaro matrix and its immediate offspring

For the Cesaro matrix C 2 B(l2) and the unilateral shift U, it is known that C and its immediate ospring UCU are both hyponormal and noncompact, and they have the same norm and the same spectrum. Here we investigate similarity and unitary equivalence for C and U*CU, as well as further generations of ospring. Necessary conditions are found for a lower triangular factorable matrix to be unitarily equivalent to its immediate ospring. A specialized result is obtained for factorable matrices having a constant main diagonal. Along the way, a more general result is also obtained: necessary conditions are found for two lower triangular factorable matrices to be unitarily equivalent.