The index locality principle in elliptic theory.

*(English. Russian original)*Zbl 1001.58017
Funct. Anal. Appl. 35, No. 2, 111-123 (2001); translation from Funkts. Anal. Prilozh. 35, No. 2, 37-52 (2001).

The authors prove index theorems for elliptic pseudodifferential operators and Fourier integral operators (quantized contact transformations) on manifolds with isolated conical singularities. Each theorem requires that the operator’s conormal symbol be symmetric only up to homotopy. The index formula involves an appropriately defined spectral flow associated with this homotopy and the index of an operator on a closed manifold (or manifolds in the Fourier integral operator case) made by deleting a neighborhood of the singular point and attaching two copies of the resulting manifold with boundary to a cylinder having the boundary as cross-section.

The localization implicit in this index formula is based on a relative index theorem for a class of abstract Fredholm operators.

The results in this paper were announced in [V. Nazaĭkinskiĭ and B. Sternin, Oper. Theory, Adv. Appl. 126, 229–237 (2001; Zbl 1001.58015)].

The localization implicit in this index formula is based on a relative index theorem for a class of abstract Fredholm operators.

The results in this paper were announced in [V. Nazaĭkinskiĭ and B. Sternin, Oper. Theory, Adv. Appl. 126, 229–237 (2001; Zbl 1001.58015)].

Reviewer: Peter Haskell (Blacksburg)

##### MSC:

58J20 | Index theory and related fixed-point theorems on manifolds |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

58J30 | Spectral flows |

19K56 | Index theory |