Chafiq Benhida

Jasang Yoon

Raúl E. Curto

Sang Hoon Lee

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 357 (2019) 799–802Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comFunctional analysisJoint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operatorsSpectres joints des transformées d’Aluthge sphériques de n-uplets commutatifs d’opérateurs d’un espace de HilbertChafiq Benhida a,1, Raúl E. Curto b,2, Sang Hoon Lee c,3, Jasang Yoon d,4a UFR de mathématiques, Université des sciences et technologies de Lille, 59655 Villeneuve-d’Ascq cedex, Franceb Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USAc Department of Mathematics, Chungnam National University, Daejeon, 34134, Republic of Koread School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 27 June 2019Accepted after revision 7 October 2019Available online 18 October 2019Presented by Gilles PisierLet T ≡ (T1, · · · , Tn) be a commuting n-tuple of operators on a Hilbert space H, and let Ti ≡ V i P (1 ≤ i ≤ n) be its canonical joint polar decomposition (i.e. P :=√T ∗1 T1 + · · · + T ∗n Tn , (V 1, · · · , Vn) a joint partial isometry, and ⋂ni=1 ker Ti =⋂ni=1 ker V i =ker P ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple T̂ := (√P V 1√P , · · · , √P Vn√P ). We prove that σT(T̂) = σT(T), where σT 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 T̂implies 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.© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoit T ≡ (T1, · · · , Tn) un n-uplet commutatif d’opérateurs sur un espace de Hilbert H, et soient Ti ≡ V i P (1 ≤ i ≤ n) sa décomposition polaire jointe canonique (i.e. P :=√T ∗1 T1 + · · · + T ∗n Tn , (V 1, · · · , Vn) une isométrie partielle jointe et ⋂ni=1 ker Ti =⋂ni=1 ker V i = ker P ). La transformée d’Aluthge sphérique de T est le n-uplet (nécessairement commutatif) T̂ := (√P V 1√P , · · · , √P Vn√P ). Nous démontrons que σT(T̂) = σT(T), où σTdésigne 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 T̂ implique l’inversibilité E-mail addresses: chafiq.benhida@univ-lille.fr (C. Benhida), raul-curto@uiowa.edu (R.E. Curto), slee@cnu.ac.kr (S.H. Lee), jasang.yoon@utrgv.edu (J. Yoon).1 The first named author was partially supported by Labex CEMPI (ANR-11-LABX-0007-01).2 The second named author was partially supported by NSF Grant DMS-1302666.3 The third named author was partially supported by NRF (Korea) grant No. 2016R1D1A1B03933776.4 The fourth named author was partially supported by a grant from the University of Texas System and the Consejo Nacional de Ciencia y Tecnología de México (CONACYT).https://doi.org/10.1016/j.crma.2019.10.0031631-073X/© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.800 C. Benhida et al. / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 799–802de P . Comme conséquence, nous pouvons étendre notre résultat à d’autres systèmes spectraux définis à partir des complexes de Koszul.© 2019 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionLet H be a complex infinite-dimensional Hilbert space, let B(H) denote the algebra of bounded linear operators on H, and let T ∈ B(H). For T ≡ V |T | the canonical polar decomposition of T , we let T̃ := |T |1/2 V |T |1/2 denote the Aluthge transform of T [1]. It is well known that T is invertible if and only if T̃ is invertible; moreover, the spectra of T and T̃ are equal. Over the last two decades, considerable attention has been given to the study of the Aluthge transform; cf. [2–4,35], [10], [13,14,17–28], [32], [39–42]). Moreover, the Aluthge transform has been generalized to the case of powers of |T | different from 12 ([5], [8], [9], [29]) and to the case of commuting pairs of operators ([13], [14]).In this note, we focus on the spherical Aluthge transform [14]. Although our results hold for arbitrary n > 2, for the reader’s convenience we will focus on the case n = 2, that is, the case of commuting pairs of Hilbert space operators. Let T ≡ (T1, T2) be a commuting pair of operators on H. We now consider the canonical polar decomposition of the column operator (T1T2); that is, (T1T2)≡(V 1V 2)P , where P := √T ∗1 T1 + T ∗2 T2 and ( V 1V 2)is a (joint) partial isometry, and subject to the constraint ⋂2i=1 ker Ti =⋂2i=1 ker V i = ker P .The spherical Aluthge transform of T is the (necessarily commuting) n-tupleT̂ := (√P V 1√P , · · · ,√P Vn√P ) ([13], [14]). (1.1)For a commuting pair T ≡ (T1, T2) of operators on H, the Koszul complex associated with T is given asK (T,H) : 0 0→ H(T1T2)−→ H⊕H (−T2 T1)−→ H 0−→ 0.Definition 1.1. A commuting pair T is said to be (Taylor) invertible if its associated Koszul complex K (T,H) is exact. The Taylor spectrum of T isσT (T) :={(λ1, λ2) ∈C2 : K ((T1 − λ1, T2 − λ2) , H) is not invertible}.The pair T is called Fredholm if each map in the Koszul complex K (T, H) has closed range and all the homology quotients are finite-dimensional. The Taylor essential spectrum isσT e(T) :={(λ1, λ2) ∈ C2 : (T1 − λ1, T2 − λ2) is not Fredholm}.J.L. Taylor showed in [36] and [37] that, if H 	= {0}, then σT (T) is a nonempty, compact subset of the polydisc of multi-radius r(T) := (r(T1), r(T2)), where r(Ti) is the spectral radius of Ti (i = 1, 2). (For additional facts about these joint spectra, the reader is referred to [11,12,15,16] and [38].)As shown in [11] and [16], the Fredholmness of T can be detected in the Calkin algebra Q(H) := B(H)/K(H). (Here K denotes the closed two-sided ideal of compact operators; we also let π : B(H) −→ Q(H) denote the quotient map.) Concretely, T is Fredholm on H if and only if the pair of left multiplication operators Lπ(T) := (Lπ(T1), Lπ(T2)) is Taylor invertible when acting on Q(H). In particular, T is left Fredholm on H if and only if Lπ(T) is bounded below on Q(H).Problem 1.2. Let T ≡ (T1, T2) be a commuting pair of operators.(i) Assume that T is (Taylor) invertible (resp. Fredholm). Is ̂T also (Taylor) invertible (resp. Fredholm)?(ii) Is the Taylor spectrum (resp. Taylor essential spectrum) of ̂T equal to that of T?We first prove that σT(̂T) = σT(T). 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 T̂ implies the invertibility of P ; P then helps to establish an isomorphism between the relevant Koszul complexes. As a consequence, we can readily extend the above result to other spectral systems that rely on the Koszul complex for their definitions, including spectral systems on Q(H).C. Benhida et al. / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 799–802 8012. Main resultsRecall the joint polar decomposition of T and the spherical Aluthge transform of T; cf. (1.1). We now state our first main result.Theorem 2.1. Assume that T or ̂T is left invertible; that is, the associated Koszul complex is exact at the left stage, and the range of the corresponding boundary map is closed. Then the operator P is invertible.Proof. Case 1. If T is left invertible, then T ∗1 T1 + T ∗2 T2 is invertible, and therefore P is invertible.Case 2. If ̂T is left invertible, then it is bounded below; that is, there exists a constant c > 0 such that∥∥∥√P V 1√P x∥∥∥2 + ∥∥∥√P V 2√P x∥∥∥2 ≥ c2 ‖x‖2 .Since (V 1, V 2) is a joint partial isometry, it readily follows that∥∥∥√P x∥∥∥2 + ∥∥∥√P x∥∥∥2 ≥ c2‖P‖ ‖x‖2 .As a result, √P is bounded below, so P is invertible. We are now ready to state our second main result.Theorem 2.2. Let T = (T1, T2) be a commuting pair of operators on H. Then,T is (Taylor) invertible ⇐⇒ T̂ is (Taylor) invertible.We now recall the notion of criss-cross commutativity.Definition 2.3. Let A ≡ (A1, · · · , An) and B ≡ (B1, · · · , Bn) be two n-tuples of operators on H. We say that A and B criss-cross commute (or that A criss-cross commutes with B) if Ai B j Ak = Ak B j Ai and Bi A j Bk = Bk A j Bi for all i, j, k = 1, · · · , n. Observe that we do not assume that A or B is commuting.Definition 2.4. Given two n-tuples A and B we define AB := (A1 B1, · · · , An Bn) and BA := (B1 A1, · · · , Bn An).Remark 2.5. It is an easy consequence of Definition 2.3 that, if A and B criss-cross commute and AB is commuting, then BAis also commuting.Lemma 2.6. Let T ≡ (T1, T2) be a commuting pair of operators on H, let P :=√T ∗1 T1 + T ∗2 T2 , and let ̂T be its spherical Aluthge transform. Then A ≡ (A1, A2) := (√P , √P ) and B ≡ (B1, B2) := (V 1√P , V 2√P ) criss-cross commute. As a consequence, ̂T(= BA)is commuting.Lemma 2.7. (cf. [6] and [7]) Let A criss-cross commute with B on H, and assume that AB is commuting. Then σT(BA) \ {0} = σT(AB) \{0}.We now prove our third main result.Theorem 2.8. Let T = (T1, T2) be a commuting pair of operators on H. ThenσT(T) = σT(̂T).Proof. Let λ ∈C2. If λ = (0, 0), use Theorem 2.2; if λ 	= (0, 0), use Lemma 2.7. Remark 2.9. (i) Theorems 2.1, 2.2 and 2.8 can be easily extended to other spectral systems whose definition is given in terms of the Koszul complex; e.g., the left k-spectral systems σπ,k defined by W. Słodkowski and W. Żelazko ([33], [34]). For, the Proof of Theorem 2.1 (which uses only left invertibility of the relevant Koszul complex) works well in case T or T̂. Once we know that √P is invertible, the Koszul complexes of T and ̂T are isomorphic, so 0 /∈ σπ,k(T) if and only if 0 /∈ σπ,k (̂T).(ii) Similarly, Theorem 2.7 admits an easy extension to Słodkowski’s left k-spectra (cf. [6], [7]), since the Proof of Theorem 2.7relies on the isomorphism of the Koszul complexes for T and ̂T, implemented by √P .(iii) On the other hand, the above results cannot be extended to Słodkowski’s right k-spectra; for, consider the adjoint U ∗+of the (unweighted) unilateral shift U+ . It is easy to see that U∗+ is onto while Û∗+ is not.802 C. Benhida et al. / C. R. Acad. Sci. Paris, Ser. I 357 (2019) 799–802Our final main result deals with Fredholmness.Theorem 2.10. Let T = (T1, T2) be a commuting pair of operators on H. ThenσT e(T) = σT e (̂T).Moreover, for each λ /∈ σT e(T), we haveind (T − λ) = ind (̂T − λ),where ind denotes the Fredholm index.Sketch of proof. In Theorem 2.1, one can replace “left invertible” for the Koszul complex with “left Fredholm” and “invert-ible” for P with “Fredholm.” A similar adjustment works for Theorems 2.2 and 2.8. In the analog of Theorem 2.2, one first proves that √P is bounded below in the orthogonal complement of ker T1 ∩ ker T2; since this kernel is finite dimensional, it follows that √P is Fredholm. In Theorem 2.8, one needs to replace Lemma 2.7 with the results for Fredholmness proved in [6], [7], [30], and [31]. 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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

Numérisation de Documents Anciens Mathématiques

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2019.10.003.pdf

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

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