Joint spectra of spherical Aluthge transforms of commuting <i>n</i>-tuples of Hilbert space operators

Benhida, Chafiq; Curto, Raúl E.; Lee, Sang Hoon; Yoon, Jasang

doi:10.1016/j.crma.2019.10.003

Functional analysis

Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators
[Spectres joints des transformées d'Aluthge sphériques de n-uplets commutatifs d'opérateurs d'un espace de Hilbert]

Présenté par : Gilles Pisier

Benhida, Chafiq ¹ ; Curto, Raúl E.² ; Lee, Sang Hoon ³ ; Yoon, Jasang ⁴

¹ UFR de mathématiques, Université des sciences et technologies de Lille, 59655 Villeneuve-d'Ascq cedex, France
² Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
³ Department of Mathematics, Chungnam National University, Daejeon, 34134, Republic of Korea
⁴ School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA

Comptes Rendus. Mathématique, Tome 357 (2019) no. 10, pp. 799-802.

Résumé
Abstract

Soit $T \equiv (T_{1}, \dots, T_{n})$ un n-uplet commutatif d'opérateurs sur un espace de Hilbert $H$ , et soient $T_{i} \equiv V_{i} P (1 \leq i \leq n)$ sa décomposition polaire jointe canonique (i.e. $P : = \sqrt{T_{1}^{⁎} T_{1} + \dots + T_{n}^{⁎} T_{n}}$ , $(V_{1}, \dots, V_{n})$ une isométrie partielle jointe et $⋂_{i = 1}^{n} \ker T_{i} = ⋂_{i = 1}^{n} \ker V_{i} = \ker P$ ). La transformée d'Aluthge sphérique de T est le n-uplet (nécessairement commutatif) $\hat{T} : = (\sqrt{P} V_{1} \sqrt{P}, \dots, \sqrt{P} V_{n} \sqrt{P})$ . Nous démontrons que $σ_{T} (\hat{T}) = σ_{T} (T)$ , où $σ_{T}$ désigne le spectre de Taylor. Nous procédons pour cela en deux étapes : en dehors de l'origine, nous utilisons les outils et les techniques de la commutativité criss-cross ; à l'origine, nous prouvons que l'inversibilité à gauche de T ou de $\hat{T}$ implique l'inversibilité de P. Comme conséquence, nous pouvons étendre notre résultat à d'autres systèmes spectraux définis à partir des complexes de Koszul.

Let $T \equiv (T_{1}, \dots, T_{n})$ be a commuting n-tuple of operators on a Hilbert space $H$ , and let $T_{i} \equiv V_{i} P (1 \leq i \leq n)$ be its canonical joint polar decomposition (i.e. $P : = \sqrt{T_{1}^{⁎} T_{1} + \dots + T_{n}^{⁎} T_{n}}$ , $(V_{1}, \dots, V_{n})$ a joint partial isometry, and $⋂_{i = 1}^{n} \ker T_{i} = ⋂_{i = 1}^{n} \ker V_{i} = \ker P$ ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple $\hat{T} : = (\sqrt{P} V_{1} \sqrt{P}, \dots, \sqrt{P} V_{n} \sqrt{P})$ . We prove that $σ_{T} (\hat{T}) = σ_{T} (T)$ , where $σ_{T}$ denotes the Taylor spectrum. We do this in two stages: away from the origin, we use tools and techniques from criss-cross commutativity; at the origin, we show that the left invertibility of T or $\hat{T}$ implies the invertibility of P. As a consequence, we can readily extend our main result to other spectral systems that rely on the Koszul complex for their definitions.

Reçu le : 2019-06-27
Accepté le : 2019-10-07
Publié le : 2019-10-01

DOI : 10.1016/j.crma.2019.10.003

Affiliations des auteurs :

Benhida, Chafiq ¹ ; Curto, Raúl E. ² ; Lee, Sang Hoon ³ ; Yoon, Jasang ⁴

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     title = {Joint spectra of spherical {Aluthge} transforms of commuting \protect\emph{n}-tuples of {Hilbert} space operators},
     journal = {Comptes Rendus. Math\'ematique},
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Benhida, Chafiq; Curto, Raúl E.; Lee, Sang Hoon; Yoon, Jasang. Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators. Comptes Rendus. Mathématique, Tome 357 (2019) no. 10, pp. 799-802. doi : 10.1016/j.crma.2019.10.003. http://www.numdam.org/articles/10.1016/j.crma.2019.10.003/

Bibliographie
Cité par

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

[2] Ando, T. Aluthge transforms and the convex hull of the spectrum of a Hilbert space operator, Recent Advances in Operator Theory and Its Applications, Operator Theory: Advances and Applications, vol. 160, 2005, pp. 21-39

[3] Ando, T.; Yamazaki, T. The iterated Aluthge transforms of a $2 \times 2$ matrix converge, Linear Algebra Appl., Volume 375 (2003), pp. 299-309

[4] Antezana, A.J.; Massey, P.; Stojanoff, D. λ-Aluthge transforms and Schatten ideals, Linear Algebra Appl., Volume 405 (2005), pp. 177-199

[5] Benhida, C. Mind Duggal transforms (Filomat, to appear) | arXiv

[6] Benhida, C.; Zerouali, E.H. On Taylor and other joint spectra for commuting n-tuples of operators, J. Math. Anal. Appl., Volume 326 (2007), pp. 521-532

[7] Benhida, C.; Zerouali, E.H. Spectral properties of commuting operations for n-tuples, Proc. Amer. Math. Soc., Volume 139 (2011), pp. 4331-4342

[8] C. Benhida, M. Chō, E. Ko, J. Lee, On the generalized mean transforms of (skew)-symmetric complex operators, preprint 2019.

[9] F. Chabbabi, Product commuting maps with the λ-Aluthge transform, preprint 2016, submitted for publication.

[10] Chō, M.; Jung, I.B.; Lee, W.Y. On Aluthge transforms of p-hyponormal operators, Integral Equ. Oper. Theory, Volume 53 (2005), pp. 321-329

[11] Curto, R. Applications of several complex variables to multi-parameter spectral theory (Conway, J.B.; Morrel, B.B., eds.), Surveys of Recent Results in Operator Theory, Vol. II, Longman Publishing Co., London, 1988, pp. 25-90

[12] Curto, R. Spectral theory of elementary operators (Mathieu, M., ed.), Elementary Operators and Applications, World Sci. Publishing, River Edge, NJ, USA, 1992, pp. 3-52

[13] Curto, R.; Yoon, J. Toral and spherical Aluthge transforms of 2-variable weighted shifts, C. R. Acad. Sci. Paris, Ser. I, Volume 354 (2016), pp. 1200-1204

[14] Curto, R.; Yoon, J. Aluthge transforms of 2-variable weighted shifts, Integral Equ. Oper. Theory, Volume 90 (2018), p. 52

[15] Curto, R.E. On the connectedness of invertible n-tuples, Indiana Univ. Math. J., Volume 29 (1980), pp. 393-406

[16] Curto, R.E. Fredholm and invertible n-tuples of operators. The deformation problem, Trans. Amer. Math. Soc., Volume 266 (1981), pp. 129-159

[17] 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

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

[19] Foiaş, C.; Jung, I.B.; Ko, E.; Pearcy, C. Complete contractivity of maps associated with the Aluthge and Duggal transforms, Pac. J. Math., Volume 209 (2003), pp. 249-259

[20] Garcia, S.R. Aluthge transforms of complex symmetric operators, Integral Equ. Oper. Theory, Volume 60 (2008), pp. 357-367

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

[22] Jung, I.B.; Ko, E.; Pearcy, C. Spectral pictures of Aluthge transforms of operators, Integral Equ. Oper. Theory, Volume 40 (2001), pp. 52-60

[23] Jung, I.B.; Ko, E.; Pearcy, C. The iterated Aluthge transform of an operator, Integral Equ. Oper. Theory, Volume 45 (2003), pp. 375-387

[24] Jung, I.B.; Ko, E.; Pearcy, C. Complete contractivity of maps associated with Aluthge and Duggal transforms, Pac. J. Math., Volume 209 (2003), pp. 249-259

[25] Kim, J.; Yoon, J. Aluthge transforms and common invariant subspaces for a commuting n-tuple of operators, Integral Equ. Oper. Theory, Volume 87 (2017), pp. 245-262

[26] Kim, M.K.; Ko, E. Some connections between an operator and its Aluthge transform, Glasg. Math. J., Volume 47 (2005), pp. 167-175

[27] Kimura, F. Analysis of non-normal operators via Aluthge transformation, Integral Equ. Oper. Theory, Volume 50 (2004), pp. 375-384

[28] 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

[29] Lee, S.H.; Lee, W.Y.; Yoon, J. The mean transform of bounded linear operators, J. Math. Anal. Appl., Volume 410 (2014), pp. 70-81

[30] Li, S. On the commuting properties of Taylor's spectrum, Chin. Sci. Bull., Volume 37 (1992), pp. 1849-1852

[31] Li, S. Taylor spectral invariance for crisscross commuting pairs on Banach spaces, Proc. Amer. Math. Soc., Volume 124 (1996), pp. 2069-2071

[32] Rion, K. Convergence properties of the Aluthge sequence of weighted shifts, Houst. J. Math., Volume 42 (2016), pp. 1217-1226

[33] Słodkowski, Z. An infinite family of joint spectra, Stud. Math., Volume 61 (1977), pp. 239-255

[34] Słodkowski, Z.; Żelazko, W. On joint spectra of commuting families of operators, Stud. Math., Volume 50 (1974), pp. 127-148

[35] Taher, B.; Rachidi, M.; Zerouali, E.H. On the Aluthge transforms of weighted shifts with moments of Fibonacci type. Application to subnormality, Integral Equ. Oper. Theory, Volume 82 (2015), pp. 287-299

[36] Taylor, J.L. A joint spectrum for several commuting operators, J. Funct. Anal., Volume 6 (1970), pp. 172-191

[37] Taylor, J.L. The analytic functional calculus for several commuting operators, Acta Math., Volume 125 (1970), pp. 1-48

[38] Vasilescu, F.-H. Analytic Functional Calculus and Spectral Decompositions, D. Reidel Publishing Co., Dordrecht–Boston–London, 1982

[39] Wu, P.Y. Numerical range of Aluthge transform of operator, Linear Algebra Appl., Volume 357 (2002), pp. 295-298

[40] Yamazaki, T. Parallelisms between Aluthge transformation and powers of operators, Acta Sci. Math. (Szeged), Volume 67 (2001), pp. 809-820

[41] Yamazaki, T. On numerical range of the Aluthge transformation, Linear Algebra Appl., Volume 341 (2002), pp. 111-117

[42] Yamazaki, T. An expression of spectral radius via Aluthge transformation, Proc. Amer. Math. Soc., Volume 130 (2002), pp. 1131-1137

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