### Abstract

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain Ω⊂R^{n} with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the *m*th powers of the resolvents of two Robin realizations is obtained, and for *m*>*n*/2-1 it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M.Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in *L*^{2}(Ω) is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space *L*^{2}(∂Ω).

Original language | English |
---|---|

Pages (from-to) | 319-337 |

Number of pages | 19 |

Journal | Journal of the London Mathematical Society |

Volume | 88 |

Issue number | 2 |

Early online date | 21 Apr 2013 |

DOIs | |

Publication status | Published - Oct 2013 |

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### Keywords

- elliptic operator
- resolvent power differences
- trace formulae
- singular values
- self-adjoint elliptic operators

### Cite this

*Journal of the London Mathematical Society*,

*88*(2), 319-337. https://doi.org/10.1112/jlms/jdt012

}

*Journal of the London Mathematical Society*, vol. 88, no. 2, pp. 319-337. https://doi.org/10.1112/jlms/jdt012

**Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators.** / Behrndt, Jussi; Langer, Matthias; Lotoreichik, Vladimir.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

AU - Behrndt, Jussi

AU - Langer, Matthias

AU - Lotoreichik, Vladimir

PY - 2013/10

Y1 - 2013/10

N2 - In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain Ω⊂Rn with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the mth powers of the resolvents of two Robin realizations is obtained, and for m>n/2-1 it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M.Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in L2(Ω) is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space L2(∂Ω).

AB - In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain Ω⊂Rn with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the mth powers of the resolvents of two Robin realizations is obtained, and for m>n/2-1 it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M.Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in L2(Ω) is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space L2(∂Ω).

KW - elliptic operator

KW - resolvent power differences

KW - trace formulae

KW - singular values

KW - self-adjoint elliptic operators

UR - http://www.scopus.com/inward/record.url?scp=84881544756&partnerID=8YFLogxK

UR - http://www.lms.ac.uk/publication/jlms

U2 - 10.1112/jlms/jdt012

DO - 10.1112/jlms/jdt012

M3 - Article

VL - 88

SP - 319

EP - 337

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 2

ER -