For free atomic Hermitian operators p and q, the boundary heuristic for the Brown measure of X = p + iq implies the boundary is an algebraic curve, and the paper provides an explicit resultant-based algorithm to produce the defining polynomial.
The Brown Measure of Non-Hermitian Sums of Projections
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abstract
We compute the Brown measure of the non-normal operators $X = p + i q$, where $p$ and $q$ are Hermitian, freely independent, and have spectra consisting of $2$ atoms. The computation relies on the model of the non-trivial part of the von Neumann algebra generated by 2 projections as $2 \times 2$ random matrices. We observe that these measures are supported on hyperbolas and note some other properties related to their atoms and symmetries.
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Quaternionic Green's Function and the Brown Measure of Atomic Operators
For free atomic Hermitian operators p and q, the boundary heuristic for the Brown measure of X = p + iq implies the boundary is an algebraic curve, and the paper provides an explicit resultant-based algorithm to produce the defining polynomial.