# Inequality involving norm of Dirac matrix

1. Sep 26, 2012

### evilcman

Hello,

I am reading a set of lecture notes on lattice QCD:
http://luscher.web.cern.ch/luscher/lectures/LesHouches09.pdf

There is an inequality here i do not understand. It is numbered (1.51)
$$|| (1-P) D^{-1} (1-P)|| \leq || (1-P) (D^{\dagger}D)^{-1} (1-P)||^{1/2}$$

|| ... || is the operator norm derived from the standard scalar product of Dirac
fields.

P is an orthogonal projector to some arbitrary subspace.

D is a discretized version of the Dirac-operator. It is assumed that D is invertible
and it's eigenvalue spectrum is contained in a disk in the right half of the
complex plane, with center M+m and radius M, m and M being positive real numbers.
Nothing else is assumed of D.

Any idea how this inequality can be derived? I don't see it...

Thanks...