On normal extensions of submatrices (Q1405046)

The extension of a bounded linear transformation from a subspace of a Hilbert space into the whole space is studied. Some sufficient conditions of subnormality for the submatrix \(\left[\begin{smallmatrix} A\\ B\end{smallmatrix}\right]\) are obtained. These conditions are deduced by means of the geometric behavior of the matrices \(A\) and \(B\). For \(B\) of rank one these sufficient conditions are also necessary for subnormality of \(\left[\begin{smallmatrix} A\\ B\end{smallmatrix}\right]\). It is also shown that \(XX^*= B^*B\) if and only if \(X^*= VB\) for some \(k\times k\) unitary matrix \(V\).