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arXiv · 2103.14325

Invariant subspaces of elliptic systems I: pseudodifferential projections

Matteo Capoferri·Dmitri Vassiliev
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Abstract

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of $m$ orthonormal pseudodifferential projections commuting with the operator $A$ and provide an algorithm for the computation of their full symbols, as well as explicit closed formulae for their subprincipal symbols. Pseudodifferential projections yield a decomposition of $L^2(M)$ into invariant subspaces under the action of $A$ modulo $C^\infty(M)$. Furthermore, they allow us to decompose $A$ into $m$ distinct sign definite pseudodifferential operators. Finally, we represent the modulus and the Heaviside function of the operator $A$ in terms of pseudodifferential projections and discuss physically meaningful examples.

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Matteo Capoferri, Dmitri Vassiliev. 2022-02-08. Invariant subspaces of elliptic systems I: pseudodifferential projections. https://doi.org/10.1016/j.jfa.2022.109402

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