Anticommutator Norm Formula for Projection Operators

TL;DR

This paper establishes a precise formula for the norm of the anticommutator of two projection operators on Hilbert space, revealing a quadratic relationship and contrasting it with the commutator norm.

Contribution

The paper proves a new exact formula for the anticommutator norm of projection operators, highlighting its quadratic dependence on the product norm and contrasting it with the commutator.

Findings

Anticommutator norm equals \\|fg\\| + \\|fg\\|^2.

Bounds for the commutator norm based on \\|fg\\|.

Contrast between commutator and anticommutator norms.

Abstract

We prove that for any two projection operators $f,g$ on Hilbert space, their anticommutator norm is given by the formula \[\|fg + gf\| = \|fg\| + \|fg\|^2.\] The result demonstrates an interesting contrast between the commutator and anticommutator of two projection operators on Hilbert space. Specifically, the norm of the anticommutator $\|fg + gf\|$ is a simple quadratic function of the norm $\|fg\|$ while the commutator norm $\|fg - gf\|$ is not a function of $\|fg\|$. Nevertheless, the result gives the following bounds that are functions of $\|fg\|$ on the commutator norm: $\|fg\| - \|fg\|^2 \le \|fg - gf\| \le \|fg\|$.