Abstract
Given a compact manifold M with boundary ∂M, in this paper we introduce a global symbolic calculus of pseudo-differential operators associated to (M,∂M). The symbols of operators with boundary conditions on ∂M are defined in terms of the biorthogonal expansions in eigenfunctions of a fixed operator L with the same boundary conditions on ∂M. The boundary ∂M is allowed to have (arbitrary) singularities. As an application, several criteria for the membership in Schatten classes on L2(M) and r-nuclearity on Lp(M) are obtained. We also describe a new addition to the Grothendieck–Lidskii formula in this setting. Examples and applications are given to operators on M=[0,1]n with non-periodic boundary conditions, and of operators with non-local boundary conditions.
Highlights
If A is an operator on a manifold M with boundary, satisfying some boundary conditions on ∂M, we establish criteria for A to belong to r-Schatten classes on L2(M ) and to be r-nuclear on Lp(M )
Given a compact manifold M with boundary ∂M, in this paper we introduce a global symbolic calculus of pseudo-differential operators associated to (M, ∂M )
We introduce the basic symbolic calculus on manifolds with boundary based on biorthogonal systems as an extension of the theory developed in [30]
Summary
If A is an operator on a manifold M with boundary, satisfying some boundary conditions on ∂M , we establish criteria for A to belong to r-Schatten classes on L2(M ) and to be r-nuclear on Lp(M ). N, in which case the global toroidal pseudo-differential calculus as developed in [29,28] can be applied The latter (periodic) setting is considerably simpler because such calculus is based on the self-adjoint operator in M (the Laplacian) with periodic boundary conditions, in particular leading to the orthonormal basis of its eigenfunctions. Unless the model operator LM (L equipped with conditions on ∂M ) is self-adjoint, we can not compose A and A∗ on their domains These observations explain why the method of studying Schatten classes via the notion of r-nuclearity on Banach spaces becomes more appropriate especially in this case of non-self-adjoint boundary value problems. In Remark 2.17 we explain how the case of spectral multipliers (in L or in L∗) is included in our setting
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