On normal extensions of submatrices

In this paper, we study the extension properties of a bounded linear transformation from a subspace of a Hilbert space into the whole space (e.g., which has a normal extension). Given an n×n normal matrix A and a k×n matrix B, k n, we obtain some sufficient conditions of subnormality for the submatrix (column matrix) by means of the geometric behavior of A and B. If, in particular, B is of rank one, we show that these sufficient conditions are also necessary for subnormality of . In order to prove these results, we establish the key lemma which says that XX*=B*B if and only if X*=VB for some k×k unitary matrix V.