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

Authors:
Ken Dykema and Hanne Schultz

Journal:
Trans. Amer. Math. Soc. **361** (2009), 6583-6593

MSC (2000):
Primary 47A05; Secondary 47B99

DOI:
https://doi.org/10.1090/S0002-9947-09-04762-X

Published electronically:
July 20, 2009

MathSciNet review:
2538606

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Aluthge transform of a Hilbert space operator , where is the polar decomposition of . We prove that the map 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 implements an automorphism of the von Neumann algebra generated by the positive part of , and we prove that the iterated Aluthge transform converges to a normal operator whose Brown measure agrees with that of (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.

**1.**A. Aluthge, `On -hyponormal operators for ,'*Integr. Equ. Oper. Theory***13**(1990), 307-315. MR**1047771 (91a:47025)****2.**J. Antezana, E.R. Pujals and D. Stojanoff, `The iterated Aluthge transforms of a matrix converge,' preprint, arXiv:0711.3727.**3.**L.G. Brown, `Lidskii's Theorem in the type II case,'*Geometric methods in operator algebras (Kyoto, 1983)*, H. Araki and E.G. Effros (eds.), Pitman Res. Notes Math. Ser.**123**, Longman Sci. Tech., 1986, pp. 1-35. MR**866489 (88d:47024)****4.**M. Cho, I. Jung and W. Lee, `On Aluthge transforms of -hyponormal operators,'*Integr. Equ. Oper. Theory*,**53**(2005), 321-329. MR**2186093 (2006m:47034)****5.**J. Elton, `Sign-embeddings of ,'*Trans. Amer. Math. Soc.***279**(1983), 113-124. MR**704605 (84g:46023)****6.**B. Fuglede and R.V. Kadison, `Determinant theory in finite factors,'*Ann. of Math. (2)***55**(1952), 520-530. MR**0052696 (14:660a)****7.**I. Jung, E. Ko and C. Pearcy, `Aluthge transforms of operators,'*Integr. Equ. Oper. Theory***37**(2000), 437-448. MR**1780122 (2001i:47035)****8.**I. Jung, E. Ko and C. Pearcy, `Spectral pictures of Aluthge transforms of operators,'*Integr. Equ. Oper. Theory***40**(2001), 52-60. MR**1829514 (2002b:47007)****9.**I. Jung, E. Ko and C. Pearcy, `The iterated Aluthge transform of an operator,'*Integr. Equ. Oper. Theory***45**(2003), 375-387. MR**1971744 (2004b:47036)****10.**U. Haagerup and H. Schultz, `Invariant subspaces for operators in a general II-factor', preprint, arXiv:math.OA/0611256.**11.**D.L. Hanson and G. Pledger, `On the mean ergodic theorem for weighted averages',*Z. Wahrscheinlichkeitstheorie verw. Geb.***13**(1969), 141-149. MR**0254214 (40:7423)****12.**K. Petersen,*Ergodic Theory*, Cambridge University Press, Cambridge, 1983. MR**833286 (87i:28002)****13.**D. Wang, `Heinz and McIntosh inequalities, Aluthge transformation and the spectral radius,'*Math. Inequal. Appl.***6**(2003), 121-124. MR**1950656 (2003k:47008)****14.**T. Yamazaki, `An expression of spectral radius via Aluthge transformation,'*Proc. Amer. Math. Soc.***130**(2002), 1131-1137. MR**1873788 (2002i:47005)**

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Additional Information

**Ken Dykema**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Email:
kdykema@math.tamu.edu

**Hanne Schultz**

Affiliation:
Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark

Email:
schultz@imada.sdu.dk

DOI:
https://doi.org/10.1090/S0002-9947-09-04762-X

Keywords:
Aluthge transform,
Brown measure,
mean ergodic theorem

Received by editor(s):
December 12, 2005

Received by editor(s) in revised form:
February 2, 2008

Published electronically:
July 20, 2009

Additional Notes:
An earlier version of this paper was distributed under the title: “On the Aluthge transform: continuity properties and Brown measure”.

The first author’s research was supported in part by NSF grant DMS–0300336.

As a student of the Ph.D. school OP-ALG-TOP-GEO, the second author was partially supported by the Danish Research Training Council and The Danish National Research Foundation.

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.