Cartesian and polar decompositions of hyponormal operators (Q793313)

For the partially isometric part and the positive part in the polar decomposition of a hyponormal operator (on a complex separable Hilbert space), we obtain respectively: Theorem 1. The singular subspace of the (unique) partially isometric part of a hyponormal operator reduces the operator itself. Theorem 2. The positive part of a pure hyponormal operator can not possess eigenvalue of finite multiplicity at the maximum of its spectrum, but it can at the minimum. In addition, we obtain a new proof for a slightly refined version of a well known structure theorem concerning Cartesian decomposition of hyponormal operators. (The refined version reads as follows: The singular subspace of the real (imaginary) part of a hyponormal operator reduces the operator itself. In fact, it is this result and its proof that motivate the formulation and proof of Theorem 1.)