In these lecture notes we introduce some of the concepts and results from microlocal analysis used in the analysis of boundary value problems for elliptic differential operators, with a special emphasis on Dirac-like operators. We first consider the problem of finding elliptic boundary conditions for the ∂̄operator on the unit disk. The rather explicit results in this special case delineate the ...

Non-commutative multi-variable versions of weighted shifts arise naturally as ‘weighted’ left creation operators acting on Fock space. We investigate the unital wot-closed algebras they generate. The unweighted case yields non-commutative analytic Toeplitz algebras. The commutant can be described in terms of weighted right creation operators when the weights satisfy a condition specific to the ...

Green’s formulas for elliptic cone differential operators are established. This is done by an accurate description of the maximal domain of an elliptic cone differential operator and its formal adjoint, thereby utilizing the concept of a discrete asymptotic type. From this description, the singular coefficients replacing the boundary traces in classical Green’s formulas are deduced. CONTENTS

We derive a spectral representation for the oblate spheroidal wave operator, which is holomorphic in the aspherical parameter Ω in a neighborhood of the real line. For real Ω, estimates are derived for all eigenvalue gaps uniformly in Ω. The proof of the gap estimates is based on detailed estimates for complex solutions of the Riccati equation. The spectral representation for complex Ω is deriv...

‎Let $\Omega$ be a bounded domain in $R^{n}$ with smooth boundary‎ ‎$\partial\Omega$‎. ‎In this article‎, ‎we will investigate the spectral‎ ‎properties of non-self adjoint elliptic differential operator\\‎ ‎$(Au)(x)=-\sum^{n}_{i,j=1}\left(\omega^{2\alpha}(x)a_{ij}(x)‎ ‎\mu(x)u'_{x_{i}}(x)\right)'_{x_{j}}$‎, ‎acting Hilbert space ‎$H=L^{2}{(\Omega)}$. Dirichlet-type boundary conditions‎. ‎Here‎...