A $\mathcal{C}^k$-Seeley-Extension-Theorem for Bastiani's Differential Calculus

Functional Calculus for Semi-Bounded Operators

We build on the work by Davies, extending the Helffer-Sjöstrand Functional Calculus domain for semi-bounded operators on Banach spaces given a priori controlled growth of the resolvents. We employ Seeley's Extension Theorem to extend smooth functions on the half line to the whole line and thus indirectly define functions of these operators.

The Spectral Mapping Theorem

We give a direct non-abstract proof of the Spectral Mapping Theorem for the Helffer-Sjöstrand functional calculus for linear operators on Banach spaces with real spectra and consequently give a new non-abstract direct proof for the Spectral Mapping Theorem for self-adjoint operators on Hilbert spaces. Our exposition is closer in spirit to the proof by explicit construction of the existence of the Functional Calculus given by Davies. We apply an extension theorem of Seeley to derive a functional calculus for semi-bounded operators.

分数次Laplace-Beltrami算子与经典Helmholtz方程的本征函数等价性

Eigenfunction equivalence for the fractional Laplace-Beltrami operator and the classical Helmholtz equation

本文建立了分数次Laplace-Beltrami算子与经典Helmholtz方程的本征函数等价性，基于Seeley的椭圆算子复幂构造与伪微分符号演算完成证明，并将其应用于固定频率逆散射问题。

R-boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators

It is shown that an elliptic scattering operator $A$ on a compact manifold with boundary with coefficients in the bounded operators of a bundle of Banach spaces of class (HT) and Pisier's property $(α)$ has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of $A$ on the scattering cotangent bundle of the manifold avoids the right half-plane. This is deduced directly from a Seeley theorem, i.e. the resolvent is represented in terms of pseudodifferential operators with R-bounded symbols, thus showing by an iteration argument the R-boundedness of $λ(A…