arXiv subjects
Publications and source records attributed to Y. Safarov.
This is a review of some coordinate-free calculi of pseudodifferential operators developed in the last years. As an application, we use a coordinate-free calculus to obtain new results on the behaviour of the spectral projections of a self-adjoint elliptic second order differential operator under perturbation of coefficients.
Our main result is a theorem saying that a bounded operator $A$ on a Hilbert space belongs to a certain set associated with its self-commutator $[A^*,A]$, provided that $A-zI$ can be approximated by invertible operators for all complex numbers $z$. The theorem remains valid in a general $C^*$-algebra of real rank zero under the assumption that $A-zI$ belong to the closure of the connected component of unity in the set of invertible elements. This result implies the Brown--Douglas--Fillmore theorem and Huaxin Lin's theorem on almost commuting matrices. Moreover, it allows us to refine the former and to extend the latter to operators of infinite rank and other norms (including the Schatten norms on the space of matrices). The proof is based on an abstract theorem, which states that a normal element of a $C^*$-algebra of real rank zero satisfying the above condition has a resolution of the identity associated with any open cover of its spectrum.
The aim of the paper is to show that L. Friedlander's results on the relation between Dirichlet and Neumann counting functions (Arch. Ration. Mech. Anal. 116, 1991) remain valid in abstract setting.
Let F be a real-valued convex function on the complex plane, and let Ps(b) be a pseudodifferential operator with symbol b. Under certain natural assumptions about properties of pseudodifferential operators, we prove that the sum of F(z) over all eigenvalues z of the operator Ps(b) counted with their multiplicities does not exceed the real part of the trace of Ps(F(b)) modulo a term of the same order as the error term in the Garding inequality.
The paper extends Birkhoff's theorem on doubly stochastic matrices to some countable families of discrete probability spaces with nonempty intersections. We join every two elements lying in the same probability space by an edge and formulate our results in terms of the obtained graph.