I am new to the field of Noncommutative Geometry.I was reading the chapter on Spectral triple from the book ‘Elements of Noncommutative Geometry’ by Gracia-Bondía,Várilly and Figueroa.Now,after reading the basics of spectral triple,I have a felling that if I perturb the Dirac operator $\mathcal{D}$ of the Spectral triple $(\mathcal{A},\mathcal{H},\mathcal{D})$ by a bounded self-adjoint operator $\mathcal{S}$,then nothing really changes in the Spectral triple.For example:

1. The Fredholm Operators corresponding to $(\mathcal{A},\mathcal{H},\mathcal{D})$ and $(\mathcal{A},\mathcal{H},\mathcal{D}+\mathcal{S})$ are homotopic.

2. The Dixmier trace $Tr_w(a|D+S|^{-p}) = Tr_w(a|D|^{-p}) \space\space \forall \space a\in \mathcal{A}$.So,are these true?If so,then how to proceed for a proof?(I have proved 1. when $\mathcal{D}$ and $\mathcal{S}$ commute.But I don’t have any Idea to proceed in general).Thanks for any help.

**Answer**

**Attribution***Source : Link , Question Author : Tim , Answer Author : Community*