[Paper Review] Schatten classes and nuclearity of boundary value problems

Julio Delgado, Michael Ruzhansky+1 more|arXiv (Cornell University)|May 9, 2015

Spectral Theory in Mathematical Physics10 references3 citations

TL;DR

This paper establishes criteria for Schatten class membership and r-nuclearity of operators on $L^2(ar{\Omega})$ and $L^p(ar{\Omega})$ via global symbolic calculus based on biorthogonal eigenfunction expansions of a reference differential operator with the same boundary conditions. The key contribution is a systematic characterization of operator ideals in terms of spectral and boundary behavior, with applications to eigenvalue asymptotics and the Grothendieck-Lidskii formula.

ABSTRACT

Given a bounded open set $\Omega$, in this paper we analyse Schatten classes and nuclearity of operators in $\Omega$ satisfying some boundary conditions on the boundary of $\Omega$. Our analysis relies on the global symbolic calculus in terms of the biorthogonal expansions in eigenfunctions of a fixed differential operator with the same boundary conditions. Several criteria for the membership in Schatten classes on $L^2(\Omega)$ and r-nuclearity on $L^p(\Omega)$ are obtained, with applications (and a new addition) to the Grothendieck-Lidskii formula and asymptotic behaviour of eigenvalues. Examples and applications are given to operators on $\Omega=(0,1)^n$ with non-periodic boundary conditions, and of operators with non-local boundary conditions.

Motivation & Objective

• To develop a global symbolic calculus for operators on bounded domains with prescribed boundary conditions.
• To characterize membership in Schatten classes and r-nuclear operators on $L^p(\Omega)$ for general boundary value problems.
• To extend the Grothendieck-Lidskii formula and analyze eigenvalue asymptotics using spectral methods.
• To analyze non-periodic and non-local boundary conditions in $\Omega = (0,1)^n$ using the proposed framework.

Proposed method

• Utilizes biorthogonal expansions in eigenfunctions of a fixed differential operator with the same boundary conditions as the target operators.
• Applies global symbolic calculus to express operators in terms of their spectral symbols and eigenfunction expansions.
• Derives trace-class and Schatten norm estimates via decay rates of eigenvalues and coefficients in the expansion.
• Establishes r-nuclearity criteria on $L^p(\Omega)$ using $L^p$-boundedness and summability of singular values.
• Applies the calculus to non-periodic and non-local boundary conditions, including in the cube $(0,1)^n$, to derive concrete estimates.
• Relies on spectral theory of elliptic operators and functional calculus to link operator ideals to boundary and spectral properties.

Experimental results

Research questions

• RQ1 What conditions ensure that a pseudodifferential operator with given boundary conditions belongs to a Schatten class on $L^2(\Omega)$?
• RQ2 How can r-nuclearity be characterized for operators on $L^p(\Omega)$ using spectral expansions and boundary conditions?
• RQ3 What is the role of biorthogonal eigenfunction expansions in constructing a global symbolic calculus for boundary value problems?
• RQ4 How do non-periodic and non-local boundary conditions affect the Schatten and nuclear properties of operators?
• RQ5 Can the Grothendieck-Lidskii formula be extended and refined using this framework for general boundary value problems?

Key findings

• The paper provides sufficient conditions for an operator to belong to the Schatten class on $L^2(\Omega)$ based on the decay of coefficients in its biorthogonal eigenfunction expansion.
• It establishes criteria for r-nuclearity on $L^p(\Omega)$ by analyzing the summability of singular values derived from spectral symbols.
• The framework yields new asymptotic estimates for eigenvalues of boundary value problems, particularly for non-periodic and non-local conditions.
• The Grothendieck-Lidskii formula is extended to include operators with non-periodic and non-local boundary conditions.
• Applications to $\Omega = (0,1)^n$ demonstrate the method’s effectiveness in concrete settings, including non-periodic and non-local cases.
• The symbolic calculus allows uniform treatment of various boundary conditions, unifying spectral and operator ideal properties.