Brown measures of unbounded operators affiliated with a finite von Neumann algebra

By Uffe Haagerup, Hanne Schultz. 2006

111 citations recorded by the source corpus.

In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite von Neumann algebra. Moreover, we compute the Brown measure of all unbounded $R$-diagonal operators in this class. As a particular case, we determine the Brown measure $z=xy^{-1}$, where $(x,y)$ is a circular system in the sense of Voiculescu, and we prove that for all $n\in \mathsf N$, $z^n\in L^p$ if and only if $0

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/13e733fb6d6c681275508cd7cce09205cee7067d

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Citations in the corpus

Around the circular law (2011, background): “Beyond bounded operators, and as explained in Brown [30] and in Haagerup and Schultz [76], it is possible to define, for a class M̄ ⊃ M of closed densely defined operators affiliated with M, a probability measure on C called the Brown spectral measure of a ∈ M̄.”
Invariant subspaces for operators in a general II1-factor (2006, background): “that N contains a circular system fx;yg which is ⁄-free from M. Moreover, we can deflne the unbounded operator z = xyi1 which by [ HS , Theorem 5.2] belongs to Lp(N;?) for”
Spectrum of Non-Hermitian Heavy Tailed Random Matrices (2010, background): “[14, 26, 30], but we will not pursue this goal here.”
The single ring theorem (2009, methodology): “R-diagonal operators were introduced by Nica and Speicher [19] in the context of free probability; they represent the weak*-limit (or more precisely, the limit in ∗-moments) of operators of the form UT with U unitary with size going to infinity and T diagonal, and were intensively studied in the las”
Free Probability and Random Matrices (2014, methodology): “For more details and, in particular, the proofs of the above used facts about Rdiagonal elements and the relation between Sx and Sx−1 one should see the original paper of Haagerup and Schultz [86].”
On the Solvability Complexity Index, the n-pseudospectrum and approximations of spectra of operators (2011, methodology): “It can be shown [31] that f is subharmonic and therefore gives rise to a measure (see Section 3 in [37])”
ENTROPY, DETERMINANTS, AND L 2 -TORSION (2012, methodology): “The Haagerup-Schultz algebra 281 6.3.”