In mathematics, the Brown measure of an operator in a finite factor is a probability measure on the complex plane which may be viewed as an analog of the spectral counting measure (based on algebraic multiplicity) of matrices.

It is named after Lawrence G. Brown.

Definition

Let ${\displaystyle {\mathcal {M))}$ be a finite factor with the canonical normalized trace ${\displaystyle \tau }$ and let ${\displaystyle I}$ be the identity operator. For every operator ${\displaystyle A\in {\mathcal {M)),}$ the function

$${\displaystyle \lambda \mapsto \tau (\log \left|A-\lambda I\right|),\;\lambda \in \mathbb {C} ,}$$

${\displaystyle \mathbb {C} }$

$${\displaystyle \mu _{A}(\mathrm {d} (a+b\mathbb {i} )):={\frac {1}{2\pi ))\nabla ^{2}\tau (\log \left|A-(a+b\mathbb {i} )I\right|)\mathrm {d} a\mathrm {d} b}$$

${\displaystyle A.}$

${\displaystyle \nabla ^{2))$

The subharmonic function can also be written in terms of the Fuglede−Kadison determinant ${\displaystyle \Delta _{FK))$ as follows

$${\displaystyle \lambda \mapsto \log \Delta _{FK}(A-\lambda I),\;\lambda \in \mathbb {C} .}$$