##
**Trace expansions and the noncommutative residue for manifolds with boundary.**
*(English)*
Zbl 0980.58017

The authors extend the theory of the trace functional on the algebra of classical pseudodifferential operators on a closed manifold to the case of pseudodifferential boundary value problems in the Boutet de Monvel calculus.

To put things in context, let us recall that, as discovered by Wodzicki, there is a trace on the algebra \(\Psi(x)\) of the classical pseudodifferential operators on a closed manifold, i.e. a linear functional \(\tau\) on \(\Psi(X)\) such that \(\tau(\varphi \psi)= \tau(\psi \varphi)\). The trace of \(A\in \Psi(x)\), called the non-commutative residue and denoted \(\text{res} (A)\), was defined by the following formula: \[ \text{res} (A) =\text{ord} P\bullet {d\over du}|_{u=0} \text{Res}_{s=1} Tr\bigl((P+uA)^{-s} \bigr). \] In this formula, \(P\) is an auxiliary invertible classical pseudodifferential operator of sufficiently large order subject to some additional condition. Then, by Seeley, complex powers \((P+uA)^{-s}\) are well defined for small \(u\) and there exists a meromorphic extension to the complex plane \(C\) of \(Tr(P+uA)^{-s}\). Then \(\text{res}(A)\) as defined above is independent of \(P\) and given by the following formula \[ \text{res} (A)=(2\pi)^{-n}\int_{S^*X} tr a_{-n}(x,\xi) d\sigma \] where \(a_{-n}(x,\xi)\) is the \(-n\)-th term in the asymptotic expansion of the symbol of \(A\).

This formula was extended by Fedosov, Golse, Leichtnam and Schrohe to operators \(A\) in the Boutet de Monvel algebra of polyhomogeneous pseudodifferential boundary value problems on a compact manifold \(X\) with boundary \(X'\) thus proving that a trace functional exists also on this algebra. Said formula is similar to the formula 2 above, but necessarily more complicated as some “boundary” expressions are to be taken into account.

Thus an interpretation of the trace on the Boutet de Monvel algebra as the residue of a meromorphic function in the style of the formula 1 above was lacking to make its theory fully parallel to that of the trace on the algebra of \(\psi\)do’s on closed manifolds. The main object of the present paper is to fill the gap. This goal is achieved by proving that for a suitable auxiliary invertible \(B\) (the Dirichlet realization of a second order strongly elliptic operator in the interior of \(X\) and an elliptic operator on the boundary \(X')\) the holomorphic function \(Tr(AB^{-s})\), \(\text{Re} s\gg 0\), has a meromorphic extension with at most double poles and the trace of \(A\) is given by the following formula: \[ \text{res} (A)=\text{ord} B\bullet\text{Res}_{s=0} Tr(AB^{-s}). \] The proof of these facts is based on beautiful expansions of trace of certain operators related to the boundary value problem \(A\) and auxiliary operator \(B\) extending earlier results of Grubb and Seeley providing analogous expansions for pseudodifferential operators on closed manifolds. In fact these expansions are lovely results in themselves and proof of them and detecting the components of \(\text{res} (A)\) in said expansions form the bulk of the paper.

To put things in context, let us recall that, as discovered by Wodzicki, there is a trace on the algebra \(\Psi(x)\) of the classical pseudodifferential operators on a closed manifold, i.e. a linear functional \(\tau\) on \(\Psi(X)\) such that \(\tau(\varphi \psi)= \tau(\psi \varphi)\). The trace of \(A\in \Psi(x)\), called the non-commutative residue and denoted \(\text{res} (A)\), was defined by the following formula: \[ \text{res} (A) =\text{ord} P\bullet {d\over du}|_{u=0} \text{Res}_{s=1} Tr\bigl((P+uA)^{-s} \bigr). \] In this formula, \(P\) is an auxiliary invertible classical pseudodifferential operator of sufficiently large order subject to some additional condition. Then, by Seeley, complex powers \((P+uA)^{-s}\) are well defined for small \(u\) and there exists a meromorphic extension to the complex plane \(C\) of \(Tr(P+uA)^{-s}\). Then \(\text{res}(A)\) as defined above is independent of \(P\) and given by the following formula \[ \text{res} (A)=(2\pi)^{-n}\int_{S^*X} tr a_{-n}(x,\xi) d\sigma \] where \(a_{-n}(x,\xi)\) is the \(-n\)-th term in the asymptotic expansion of the symbol of \(A\).

This formula was extended by Fedosov, Golse, Leichtnam and Schrohe to operators \(A\) in the Boutet de Monvel algebra of polyhomogeneous pseudodifferential boundary value problems on a compact manifold \(X\) with boundary \(X'\) thus proving that a trace functional exists also on this algebra. Said formula is similar to the formula 2 above, but necessarily more complicated as some “boundary” expressions are to be taken into account.

Thus an interpretation of the trace on the Boutet de Monvel algebra as the residue of a meromorphic function in the style of the formula 1 above was lacking to make its theory fully parallel to that of the trace on the algebra of \(\psi\)do’s on closed manifolds. The main object of the present paper is to fill the gap. This goal is achieved by proving that for a suitable auxiliary invertible \(B\) (the Dirichlet realization of a second order strongly elliptic operator in the interior of \(X\) and an elliptic operator on the boundary \(X')\) the holomorphic function \(Tr(AB^{-s})\), \(\text{Re} s\gg 0\), has a meromorphic extension with at most double poles and the trace of \(A\) is given by the following formula: \[ \text{res} (A)=\text{ord} B\bullet\text{Res}_{s=0} Tr(AB^{-s}). \] The proof of these facts is based on beautiful expansions of trace of certain operators related to the boundary value problem \(A\) and auxiliary operator \(B\) extending earlier results of Grubb and Seeley providing analogous expansions for pseudodifferential operators on closed manifolds. In fact these expansions are lovely results in themselves and proof of them and detecting the components of \(\text{res} (A)\) in said expansions form the bulk of the paper.

Reviewer: W.Oledzki (Warszawa)

### MSC:

58J42 | Noncommutative global analysis, noncommutative residues |

58J32 | Boundary value problems on manifolds |

### Keywords:

noncommutative residue; trace expansions; pseudodifferential boundary value problems; Boutet de Monvel calculus### References:

[1] | Invent. Math. 50 pp 219– (1979) |

[2] | Acta Math. 126 pp 11– (1971) |

[3] | Fedosov B. V., J. Funct. Anal. 142 pp 1– (1996) |

[4] | Isr. J. Math. 10 pp 32– (1971) |

[5] | G. Grubb, Heat operator trace expansions and index for general Atiyah-Patodi-Singer problems, Comm. P. D. E. 17 (1992), 2031-2077. · Zbl 0773.58025 |

[6] | G. Grubb, Functional calculus of pseudodi erential boundary problems, Progr. Math. 65, second edition, BirkhaEuser, Boston 1996, rst edition 1986. |

[7] | Ark. Mat. 37 pp 45– (1999) |

[8] | Grubb G., Acta Math. 171 pp 165– (1993) |

[9] | Grubb G., Invent. Math. 121 pp 481– (1995) |

[10] | Grubb G., J. Geom. An. 6 pp 31– (1996) |

[11] | Adv. Math. 102 pp 184– (1985) |

[12] | C. Kassel, Le reAsidu non commutatif [d’apreAs M. Wodzicki], AsteArisque 177-178 (1989), 199-229; SeAm.Bourbaki, 41eAme anneAe41 (1988-99). |

[13] | Ann. Global Anal. Geom. 17 pp 151– (1999) |

[14] | J. Sov. Math. 11 pp 1– (1979) |

[15] | R., Amer. Math. Soc. Proc. Symp. Pure Math. 10 pp 288– (1967) |

[16] | R., Roma 1969 pp 169– |

[17] | M. Wodzicki, Spectral asymmetry and noncommutative residue (in Russian), Thesis, Stekhlov Institute of Mathematics, Moscow 1984. Department of Mathematics, University of Copenhagen.Universitetsparken5, 2100 Copenhagen, Denmark · Zbl 0538.58038 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.