arXiv subjects

Hanne Schultz

Publications and source records attributed to Hanne Schultz.

7 recordsLinked to original sources

arXiv2008

Brown measure and iterates of the Aluthge transform for some operators arising from measurable actions

We consider the Aluthge transform $|T|^{1/2}U|T|^{1/2}$ of a Hilbert space operator $T$, where $T=U|T|$ is the polar decomposition of $T$. We prove that the map that sends $T$ to its Aluthge transform is continuous with respect to the norm topology and with respect to the $*$--SOT topology on bounded sets. We consider the special case in a tracial von Neumann algebra when $U$ implements an automorphism of the von Neumann algebra generated by the positive part $|T|$ of $T$, and we prove that the iterated Aluthge transform converges to a normal operator whose Brown measure agrees with that of $T$ (and we compute this Brown measure). This proof relies on a theorem that is an analogue of von Neumann's mean ergodic theorem, but for sums weighted by binomial coefficients.

math.OA↗

arXiv2006

Invariant Subspaces for Operators in a General II_1-factor

It is shown that to every operator T in a general von Neumann factor M of type II_1 and to every Borel set B in the complex plane, one can associate a largest, closed, T-invariant subspace, K = K_T(B), affiliated with M, such that the Brown measure of T|_K is concentrated on B. Moreover, K is T-hyperinvariant, and the Brown measure of (1-P_K)T|_(1-P_K)(H) is concentrated on C\B. In particular, if T has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T in M, the limit A=\lim_{n\to\infty}[(T^n)* T^n]^{1/2n} exists in the strong operator topology and K_T(\bar{B(0,r)})=1_{[0,r]}(A), r>0.

math.OA↗

arXiv2006

Brown Measures of Unbounded Operators Affiliated with a Finite von Neumann Algebra

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 of z=xy^{-1}, where (x,y) is a circular system in the sense of Voiculescu, and we prove that for all positive integers n, z^n is in L^p(M) iff 0<p< 2/(n+1).

math.OA↗

arXiv2006

Semicircularity, Gaussianity and Monotonicity of Entropy

S. Artstein, K. Ball, F. Barthe, and A. Naor have shown that if (X_j) are i.i.d. random variables, then the entropy of n^{-1/2}(X_1+....+X_n) increases as n increases. The free analogue was recently proven by D. Shlyakhtenko. That is, if (x_j) are freely independent, identically distributed, self-adjoint elements in a noncommutative probability space, then the free entropy of n^{-1/2}(x_1+....+x_n) increases as n increases. In this paper we prove that if X_1 (x_1, resp.) has finite entropy (free entropy, resp.), and if the entropy (the free entropy, resp.) is not a strictly increasing function of n, then X_1 (x_1, resp.) must be Gaussian (semicircular, resp.).

math.OA↗

arXiv2005

Brown measures of sets of commuting operators in a type II_1 factor

Using the spectral subspaces obtained in [HS], Brown's results on the Brown measure of an operator in a type II_1 factor (M,tr) are generalized to finite sets of commuting operators in M. It is shown that whenever T_1,..., T_n in M are mutually commuting operators, there exists one and only one compactly supported Borel probability measure mu_{T_1,..., T_n} on C^n such that for all alpha_1,..., alpha_n in C, tr(log|alpha_1 T_1+ ... + alpha_n T_n - 1|) is the integral of log|alpha_1 z_1 + ... + alpha_n z_n-1| w.r.t mu_{T_1,...,T_n}. Moreover, for every polynomial q in n commuting variables, mu_{q(T_1,..., T_n)} is the push-forward measure of mu_{T_1,...,T_n} via the map q. In addition it is shown that, as in [HS], for every Borelset B in C^n there is a maximal closed T_1-,..., T_n-invariant subspace K affiliated with M, such that mu_{T_1|_K,..., T_n|_K} is concentrated on B. Moreover, tr(P_K)=mu_{T_1,...,T_n}(B). This generalizes the main result from [HS] to n-tuples of commuting operators in M.

math.OA↗

arXiv2004

A Random Matrix Approach to the Lack of Projections in C*_red(F_2)

In 1982 Pimsner and Voiculescu computed the K_0- and K_1-groups of the reduced group C*-algebra C*_red(F_k) of the free group F_k on k generators and settled thereby a long standing conjecture: C*_red(F_k) has no projections except for the trivial projections 0 and 1. Later simpler proofs of this conjecture were found by methods from K-theory or from non-commutative differential geometry. In this paper we provide a new proof of the fact that C*_red(F_k) is projectionless. The new proof is based on random matrices and is obtained by a refinement of the methods recently used by the first and the third named author to show that the semigroup Ext(C*_red(F_k)) is not a group for k >= 2. By the same type of methods we also obtain that two phenomena proved by Bai and Silverstein for certain classes of random matrices: ``no eigenvalues outside (a small neighbourhood of) the support of the limiting distribution'' and ``exact separation of eigenvalues by gaps in the limiting distribution'' also hold for arbitrary non-commutative selfadjoint polynomials of independent GUE, GOE or GSE random matrices with matrix coefficients.

math.OA↗

arXiv2003

Non-commutative Polynomials of Independent Gaussian Random Matrices. The Real and Symplectic Cases

In their paper, "A new application of random matrices: Ext(C*_red(F_2)) is not a group", Haagerup and Thorbjornsen prove an extension of Voiculescu's random matrix model for independent complex self-adjoint Gaussian random matrices. We generalize their result to random matrices with real or symplectic entries (the GOE- and the GSE-ensembles) and random matrix ensembles related to these.

math.OA↗