A unitary invariant for hyponormal operators
Let \({\mathcal H}\) be a complex separable Hilbert space, let \({\mathcal L}({\mathcal H})\) denote the algebra of all bounded linear operators on \({\mathcal H}\), and let T denote a hyponormal operator on \({\mathcal H}\) (i.e. \(D:=T\) *T-TT *\(\geq 0)\). Then for any \(z\in {\mathbb{C}}\) in \({\mathcal L}({\mathcal H})\) there exists a uniquely defined contraction C(z) in \({\mathcal L}({\mathcal H})\) such that (T-z) \(*C(z)=D^{1/2}\) and C *(z)\(| Ker(T-z)\) \(*=0\). This family of contractions is the main tool in this nicely readable paper. Generalizing a result of \textit{K. F. Clancey}'s [J. Oper. Theory 13, 265- 289 (1985; Zbl 0588.47043)] it is shown that T is pure (i.e. there exists no reducing subspace on which the restriction is normal) if and only if the span of all vectors C(z)x with z in \({\mathbb{C}}\) and x in \({\mathcal X}:=Ran D^{1/2}\) is dense in \({\mathcal H}\). The germ at infinity of the \({\mathcal L}({\mathcal H})\)-valued function C *C as well as the distribution \(- D^{1/2}\partial C\) in \({\mathcal D}'({\mathbb{C}}){\hat \otimes}{\mathcal L}({\mathcal X})\) (\(\partial C\) is the complex derivative of C in the sense of distributions) are exhibited to be complete unitary invariants. Originated by \textit{Xia}'s first representation of hyponormal operators as singular integral operators [Chinese J. Math. 3, 232-246 (1963)] a two- dimensional singular-integral model is developed. Finally those positive definite kernels in \({\mathcal L}^{\infty}({\mathbb{C}}\) 2)\({\hat \otimes}_{\epsilon}{\mathcal L}({\mathcal X})\) are characterized which produce pure hyponormal operators.
scientific article; zbMATH DE number 4034459
scientific article