Functional analysis
Toral and spherical Aluthge transforms of 2-variable weighted shifts
Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1200-1204.

We introduce two natural notions of Aluthge transforms (toral and spherical) for 2-variable weighted shifts and study their basic properties. Next, we study the class of spherically quasinormal 2-variable weighted shifts, which are the fixed points for the spherical Aluthge transform. Finally, we briefly discuss the relation between spherically quasinormal and spherically isometric 2-variable weighted shifts.

Nous introduisons deux notions naturelles des transformations d'Aluthge (torales et sphériques) pour les shifts pondérés à deux variables et nous étudions leurs propriétés. Ensuite, nous étudions la classe de shifts pondérés à deux variables sphériques et quasi-normaux, qui sont les points fixes pour la transformation d'Aluthge sphérique. Enfin, nous discutons brièvement la relation entre les shifts pondérés à deux variables qui sont sphériquement quasinormaux et ceux qui sont sphériquement isométriques.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.10.005
Curto, Raúl E. 1; Yoon, Jasang 2

1 Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
2 School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA
@article{CRMATH_2016__354_12_1200_0,
     author = {Curto, Ra\'ul E. and Yoon, Jasang},
     title = {Toral and spherical {Aluthge} transforms of 2-variable weighted shifts},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1200--1204},
     publisher = {Elsevier},
     volume = {354},
     number = {12},
     year = {2016},
     doi = {10.1016/j.crma.2016.10.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2016.10.005/}
}
TY  - JOUR
AU  - Curto, Raúl E.
AU  - Yoon, Jasang
TI  - Toral and spherical Aluthge transforms of 2-variable weighted shifts
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 1200
EP  - 1204
VL  - 354
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2016.10.005/
DO  - 10.1016/j.crma.2016.10.005
LA  - en
ID  - CRMATH_2016__354_12_1200_0
ER  - 
%0 Journal Article
%A Curto, Raúl E.
%A Yoon, Jasang
%T Toral and spherical Aluthge transforms of 2-variable weighted shifts
%J Comptes Rendus. Mathématique
%D 2016
%P 1200-1204
%V 354
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2016.10.005/
%R 10.1016/j.crma.2016.10.005
%G en
%F CRMATH_2016__354_12_1200_0
Curto, Raúl E.; Yoon, Jasang. Toral and spherical Aluthge transforms of 2-variable weighted shifts. Comptes Rendus. Mathématique, Volume 354 (2016) no. 12, pp. 1200-1204. doi : 10.1016/j.crma.2016.10.005. http://www.numdam.org/articles/10.1016/j.crma.2016.10.005/

[1] Aluthge, A. On p-hyponormal operators for 0<p<1, Integral Equ. Oper. Theory, Volume 13 (1990), pp. 307-315

[2] Athavale, A. On the intertwining of joint isometries, J. Oper. Theory, Volume 23 (1990), pp. 339-350

[3] Athavale, A.; Poddar, S. On the reflexivity of certain operator tuples, Acta Math. Sci. (Szeged), Volume 81 (2015), pp. 285-291

[4] Attele, K.R.M.; Lubin, A.R. Commutant lifting for jointly isometric operators—a geometrical approach, J. Funct. Anal., Volume 140 (1996), pp. 300-311

[5] Chavan, S.; Sholapurkar, V. Rigidity theorems for spherical hyperexpansions, Complex Anal. Oper. Theory, Volume 7 (2013), pp. 1545-1568

[6] Curto, R.; Fialkow, L. Recursively generated weighted shifts and the subnormal completion problem, Integral Equ. Oper. Theory, Volume 17 (1993), pp. 202-246

[7] Curto, R.; Lee, S.H.; Yoon, J. k-hyponormality of multivariable weighted shifts, J. Funct. Anal., Volume 229 (2005), pp. 462-480

[8] Curto, R.; Lee, S.H.; Yoon, J. Subnormality of 2-variable weighted shifts with diagonal core, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013), pp. 203-207

[9] Curto, R.; Yoon, J. Jointly hyponormal pairs of subnormal operators need not be jointly subnormal, Trans. Amer. Math. Soc., Volume 358 (2006), pp. 5139-5159

[10] Dykema, K.; Schultz, H. Brown measure and iterates of the Aluthge transform for some operators arising from measurable actions, Trans. Amer. Math. Soc., Volume 361 (2009), pp. 6583-6593

[11] Eschmeier, J.; Putinar, M. Some remarks on spherical isometries, Oper. Theory, Adv. Appl., Volume 129 (2001), pp. 271-291

[12] Exner, G.R. Aluthge transforms and n-contractivity of weighted shifts, J. Oper. Theory, Volume 61 (2009), pp. 419-438

[13] Gleason, J. Quasinormality of Toeplitz tuples with analytic symbols, Houst. J. Math., Volume 32 (2006), pp. 293-298

[14] J. Gleason, Quasinormality and commuting tuples, preprint, 2004.

[15] Jung, I.; Ko, E.; Pearcy, C. Aluthge transform of operators, Integral Equ. Oper. Theory, Volume 37 (2000), pp. 437-448

[16] Kim, J.; Yoon, J. Schur product techniques for the subnormality of commuting 2-variable weighted shifts, Linear Algebra Appl., Volume 453 (2014), pp. 174-191

[17] Lee, S.H.; Lee, W.Y.; Yoon, J. Subnormality of Aluthge transform of weighted shifts, Integral Equ. Oper. Theory, Volume 72 (2012), pp. 241-251

[18] Research Wolfram, Inc., Mathematica, Version 9.0, Champaign, IL, 2013.

Cited by Sources:

The first author of this paper was partially supported by NSF Grant DMS-1302666.