On utilizing the spectral representation of selfadjoint operators in Hilbert spaces, some approximations for the n-time di¤erentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor’s type expansions are given. 1. Introduction Let U be a selfadjoint operator on the complex Hilbert space (H; h:; :i) with the spectrum Sp (U) included in the interval [m;M ] for some real n...

This is an unedited transscript of the handwritten notes of my talk at the Master Class.

We consider elliptic operators A on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of A through an observation, with an exponential cost. Following the strategy of...

A general construction of transmutation operators is developed for selfadjoint operators in Gelfand triples. Theorems regarding analyticity of generalized eigenfunctions and Paley-Wiener properties are proved.

In this paper we investigate discrete spectrum of the non-selfadjoint matrix Sturm-Liouville operator L generated in L (R+, S) by the differential expression ` (y) = −y +Q (x) y , x ∈ R+ : [0,∞) , and the boundary condition y′ (0)− ( β0 + β1λ+ β2λ 2 ) y (0) = 0 where Q is a non-selfadjoint matrix valued function. Also using the uniqueness theorem of analytic functions we prove that L has a fini...

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property 13 of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators 15 that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the...

Singular boundary conditions are formulated for non-selfadjoint Sturm-Liouville operators with singularities and turning points. For boundary value problems with singular boundary conditions properties of the spectrum are studied and the completeness of the system of root functions is proved.

For a class of bounded random selfadjoint operators (Lu)(n) =

A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trac...