For a wide class of pairs of unbounded selfadjoint operators (A,B) with bounded commutator and with operator (1+A2+B2)−1 being compact we construct aK-theoretical integer invariant ω(A,B), which is continuous, is equal to zero for commuting operators and is equal to one for the pair (x,−i d/dx). Such invariants in the case of bounded operators were constructed by T. Loring [1, 3] for matrices r...

Among all linear operators on Hilbert spaces, the compact ones (defined below) are the simplest, and most closely imitate finite-dimensional operator theory. In addition, compact operators are important in practice. We prove a spectral theorem for self-adjoint compact operators, which does not use broader discussions of properties of spectra, only using the Cauchy-Schwarz-Bunyakowsky inequality...

A general integral formula for the spectral flow of a path of unbounded selfadjoint Fredholm operators subject to certain summability conditions is derived from the interpretation of the spectral flow as a winding number.

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...

We study Voiculescu’s microstate free entropy for a single non–selfadjoint random variable. The main result is that certain additional constraints on eigenvalues of microstates do not change the free entropy. Our tool is the method of random regularization of Brown measure which was studied recently by Haagerup and the author. As a simple application we present an upper bound for the free entro...