If X,Y, Z are finite sets of selfadjoint elements in a tracial von Neumann algebra and X generates a hyperfinite von Neumann algebra, then δ0(X ∪Y ∪Z) ≤ δ0(X ∪Y )+δ0(X ∪Z)−δ0(X). We draw several corollaries from this inequality. In [9] Voiculescu describes the role of entropy in free probability. He discusses several problems in the area, one of which is the free entropy dimension problem. Free...

If a nonnegative selfadjoint linear relation A in Hilbert space and closed subspace S are assumed to satisfy that the domain of is invariant under orthogonal projector onto S, then admits particular matrix representation with respect decomposition S⊕S⊥. This used give explicit formulae for Schur complement on as well S-compression A.

We consider a general nonselfadjoint second order scalar operator in planar domain such that one of the boundary components is obtained by an arbitrary irregular curving under assumption amplitude small. The Dirichlet or Neumann condition imposed on perturbed component, and remaining part boundary. prove norm resolvent convergence operators, estimate rate two norms, spectrum operators. In selfa...