A decomposition theorem of operators for variegations (Q1326596)

Let \(B(H)\) be the set of all bounded linear operators on a Hilbert space \(H\). Let \(\mathcal P\) be a class of operators \(T\in B(H)\) with some special property, e.g., the property that \(T^* T- TT^*= 0\). \(\mathcal P\) is called a variegation if it is closed under direct sums, images of *- homomorphisms, and suboperators, i.e., restrictions to non-zero reducing subspaces. An example of a variegation would be the class of all operators \(T\) with \(e^ T- 1= 0\). Suppose that for a given operator \(T\) there exists a largest reducing subspace \(M\neq \{0\}\) such that \(T|_ M\in {\mathcal P}\). Then, \(T|_ M\) is called the \({\mathcal P}\)- part of \(T\). If there exists no non-zero reducing subspace \(M\) of \(H\) such that \(T|_ M\in {\mathcal P}\), then \(T\) is called completely non- \({\mathcal P}\), and \(T\) has no \({\mathcal P}\)-part. The main theorem proved in this paper states the following: Suppose that a class \(\mathcal P\) of operators is a variegation. Then, any \(T\in B(H)\) can be decomposed uniquely in one of the following ways: (1) \(T\in {\mathcal P}\) and the completely non-\({\mathcal P}\) part of \(T\) does not exist; (2) \(T\) is completely non-\({\mathcal P}\) and the \({\mathcal P}\)-part of \(T\) does not exist; (3) there exists a unique reducing subspace \(M\neq \{0\}\) with \(M^ \perp\neq \{0\}\) such that \(T|_ M\) is the \({\mathcal P}\)-part of \(T\) and \(T|_{M^ \perp}\) is the completely non-\({\mathcal P}\) part of \(T\). Moreover, the projection \(P_ M\) of \(H\) onto \(M\) is in the center of the von Neumann algebra \(R(T)\) generated by \(T\). Also proved is a characterization of variegation which is analogous to a theorem of G. Birkhoff on algebras and varieties.