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

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

is a

subharmonic function and its

Laplacian in the

distributional sense is a probability measure on

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

which is called the Brown measure of

$A.$ Here the Laplace operator

$\nabla ^{2}$ is complex.

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

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

See also
Direct integral – generalization of the concept of direct sumPages displaying wikidata descriptions as a fallback
References

Brown, Lawrence (1986), "Lidskii's theorem in the type $II$ case", Pitman Res. Notes Math. Ser. , Longman Sci. Tech., Harlow, 123 : 1–35 . Geometric methods in operator algebras (Kyoto, 1983).

Basic concepts Main results Special Elements/Operators
Spectrum Decomposition Spectral Theorem Special algebras Finite-Dimensional Generalizations Miscellaneous Examples Applications