Invariant Subspaces of Operators on a Hilbert Space

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In year 2006 the author proposed an approach to the invariant subspace problem for an operator on a Hilbert space, based on projection-convex combinations in $$C^{*}$$ -algebras with the unitary factorization property. In this paper, we present an operator inequality characterizing the invariant subspace of such an operator. Eight corollaries are obtained. For an operator $$C^{*}$$ -algebra $$\mathcal{A}$$ with a faithful trace, we give a sufficient condition of commutation for a partial isometry from $$\mathcal{A}$$ with a projection onto its invariant subspace.