Intersection theory and the Horn inequalities for invariant subspaces

We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_{0}, of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where `inequality' is replaced by `divisibility'. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. The result is true even for operators acting on nonseparable Hilbert spaces. For such operators the usual Horn inequalities are supplemented so as to apply to all the Jordan blocks in the model

Free evolution on algebras with two states II

Denote by $J$ the operator of coefficient stripping. We show that for any free convolution semigroup of measures $\nu_t$ with finite variance, applying a single stripping produces semicircular evolution with non-zero initial condition, $J[\nu_t] = \rho \boxplus \sigma^{\boxplus t}$, where $\sigma$ is the semicircular distribution with mean $\beta$ and variance $\gamma$. For more general freely infinitely divisible distributions $\tau$, expressions of the form $\rho \boxplus \tau^{\boxplus t}$ arise from stripping $\mu_t$, where the pairs $(\mu_t, \nu_t)$ form a semigroup under the operation of two-state free convolution. The converse to this statement holds in the algebraic setting. Numerous examples illustrating these constructions are computed. Additional results include the formula for generators of such semigroups.Comment: Numerous statements clarified following suggestions by the refere