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Journal articles on the topic 'Tuples of Operators'

Author: Grafiati

Published: 6 September 2023

Last updated: 31 July 2025

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1

Wang, Wei, Yonglu Shu, and Xingzhong Wang. "S-Mixing Tuple of Operators on Banach Spaces." Journal of Function Spaces 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/9251672.

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We consider the question: what is the appropriate formulation of Godefroy-Shapiro criterion for tuples of operators? We also introduce a new notion about tuples of operators,S-mixing, which lies between mixing and weakly mixing. We also obtain a sufficient condition to ensure a tuple of operators to beS-mixing. Moreover, we study some new properties ofS-mixing operators on several concrete Banach spaces.

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2

Hoffmann, Philipp H. W., and Michael Mackey. "(m, p)-isometric and (m, ∞)-isometric operator tuples on normed spaces." Asian-European Journal of Mathematics 08, no. 02 (2015): 1550022. http://dx.doi.org/10.1142/s1793557115500229.

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We generalize the notion of m-isometric operator tuples on Hilbert spaces in a natural way to operator tuples on normed spaces. This is done by defining a tuple analogue of (m, p)-isometric operators, so-called (m, p)-isometric operator tuples. We then extend this definition further by introducing (m, ∞)-isometric operator tuples and study properties of and relations between these objects.

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3

Ahmed, Ahmed. "Higher dimensional [m,C]-isometric commuting d-tuple of operators." Filomat 36, no. 12 (2022): 4173–84. http://dx.doi.org/10.2298/fil2212173a.

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In this paper we recover an [m,C]-isometric operators and (m,C)-isometric commuting tuples of operators on a Hilbert space studied respectively in [11] and [16], we introduce the class of [m,C]-isometries for tuple of commuting operators. This is a generalization of the class of [m,C]-isometric commuting operators on a Hilbert spaces. A commuting tuples of operators S = (S1,..., Sp) ? B(H)p is said to be [m,C]-isometric p-tuple of commuting operators if ?m (S,C):= ?m j=0 (?1)m?j (m j) (? |?|=j j!/?! CS?CS?)=0 for some positive integer m and some conjugation C. We consider a multi-variable gene

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4

Mićić, Jadranka, Zlatko Pavić, and Josip Pečarić. "Extension of Jensen's Inequality for Operators without Operator Convexity." Abstract and Applied Analysis 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/358981.

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We give an extension of Jensen's inequality for -tuples of self-adjoint operators, unital -tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.

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5

Duggal, B. P. "Subspace gaps and Weyl's theorem for an elementary operator." International Journal of Mathematics and Mathematical Sciences 2005, no. 3 (2005): 465–74. http://dx.doi.org/10.1155/ijmms.2005.465.

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A range-kernal orthogonality property is established for the elementary operatorsℰ(X)=∑i=1nAiXBiandℰ*(X)=∑i=1nAi*XBi*, whereA=(A1,A2,…,An)andB=(B1,B2,…,Bn)aren-tuples of mutually commuting scalar operators (in the sense of Dunford) in the algebraB(H)of operators on a Hilbert spaceH. It is proved that the operatorℰsatisfies Weyl's theorem in the case in whichAandBaren-tuples of mutually commuting generalized scalar operators.

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6

Charlesworth, Ian, Ken Dykema, Fedor Sukochev, and Dmitriy Zanin. "Simultaneous Upper Triangular Forms for Commuting Operators in a Finite von Neumann Algebra." Canadian Journal of Mathematics 72, no. 5 (2019): 1188–245. http://dx.doi.org/10.4153/s0008414x19000282.

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AbstractThe joint Brown measure and joint Haagerup–Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved, and the joint Brown measure and joint Haagerup–Schultz projections are shown to behave well under the Arens

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7

Aljawi, Salma, Kais Feki, and Hranislav Stanković. "Jointly $ A $-hyponormal $ m $-tuple of commuting operators and related results." AIMS Mathematics 9, no. 11 (2024): 30348–63. http://dx.doi.org/10.3934/math.20241464.

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<p>In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $ A $ on a complex Hilbert space $ \mathcal{X} $, which is called jointly $ A $-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $ m $-tuples of operators that admit adjoint operators with respect to $ A $. Mainly, we prove that if $ \mathbf{B} = (B_1, \cdots, B_m) $ is a jointly $ A $-hyponormal $ m $-tuple of commuting operators, then $ \mathbf{B} $ is jointly $ A $-normaloid. This result allows us to es

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8

Bhatia, R., and L. Elsner. "On Joint Eigenvalues of Commuting Matrices." Canadian Mathematical Bulletin 39, no. 2 (1996): 164–68. http://dx.doi.org/10.4153/cmb-1996-020-6.

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AbstractA spectral radius formula for commuting tuples of operators has been proved in recent years. We obtain an analog for all the joint eigenvalues of a commuting tuple of matrices. For a single matrix this reduces to an old result of Yamamoto.

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9

Chō, Muneo. "Joint spectra of commuting normal operators on Banach spaces." Glasgow Mathematical Journal 30, no. 3 (1988): 339–45. http://dx.doi.org/10.1017/s0017089500007436.

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The joint spectrum for a commuting n-tuple in functional analysis has its origin in functional calculus which appeared in J. L. Taylor's epoch-making paper [19] in 1970. Since then, many papers have been published on commuting n-tuples of operators on Hilbert spaces (for example, [3], [4], [5], [8], [9], [10], [21], [22]).

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10

Works, Karen, and Elke A. Rundensteiner. "Preferential Resource Allocation in Stream Processing Systems." International Journal of Cooperative Information Systems 23, no. 04 (2014): 1450006. http://dx.doi.org/10.1142/s0218843014500063.

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Overloaded data stream management systems (DSMS) cannot process all tuples within their response time. For some DSMS it is crucial to allocate the precious resources to process the most significant tuples. Prior work has applied shedding and spilling to permanently drop or temporarily place to disk insignificant tuples. However neither approach considers that tuple significance can be multi-tiered nor that significance determination can be costly. These approaches consider all tuples not dropped to be equally significant. Unlike these prior works, we take a fresh stance by pulling the most sig

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11

Hong, Kijae, Kyoungmin Kim, Young-Koo Lee, Yang-Sae Moon, Sourav S. Bhowmick, and Wook-Shin Han. "Themis: A GPU-Accelerated Relational Query Execution Engine." Proceedings of the VLDB Endowment 18, no. 2 (2024): 426–38. https://doi.org/10.14778/3705829.3705856.

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GPU-accelerated relational query execution engines have parallelized the execution of a pipeline, a sequence of operators. For the parallelization, the engines evenly partition the tuples in a table that will be scanned by the pipeline's first operator (a scan), and each thread executes the pipeline for the tuples in a partition. However, this approach leads to load imbalances since an operator returns a varying number of output tuples per input tuple, particularly under non-uniform data distributions such as skewed join key values. The load imbalances are classified into intra- and inter-warp

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12

Beinane, Sid Ahmed Ould, and Sid Ahmed Ould Ahmed Mahmoud. "On $ (n_1, \cdots, n_m) $-hyponormal tuples of Hilbert space operators." AIMS Mathematics 9, no. 10 (2024): 27784–96. http://dx.doi.org/10.3934/math.20241349.

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<p>This paper introduces a new class of multivariable operators called $ (n_1, \cdots, n_m) $-hyponormal tuples, which combine joint normal and joint hyponormal operators. A tuple of operators $ \mathcal{Q} = (\mathcal{Q}_1, \; \cdots, \mathcal{Q}_m) $ is said to be an $ (n_1, \cdots, n_m) $-hyponormal tuple for some $ (n_1, \cdots, n_m)\in \mathbb{N}^m $ if</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits_{1\leq k,\;l\leq m}\big\langle[\mathcal{Q}_k^{*n_k}, \;\mathcal{Q}_l^{n_l}]\omega_k\mid \omega_l\big\rangle\geq 0, \quad \f

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13

Altwaijry, Najla, Kais Feki, and Nicuşor Minculete. "A New Seminorm for d-Tuples of A-Bounded Operators and Their Applications." Mathematics 11, no. 3 (2023): 685. http://dx.doi.org/10.3390/math11030685.

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The aim of this paper was to introduce and investigate a new seminorm of operator tuples on a complex Hilbert space H when an additional semi-inner product structure defined by a positive (semi-definite) operator A on H is considered. We prove the equality between this new seminorm and the well-known A-joint seminorm in the case of A-doubly-commuting tuples of A-hyponormal operators. This study is an extension of a well-known result in [Results Math 75, 93(2020)] and allows us to show that the following equalities rA(T)=ωA(T)=∥T∥A hold for every A-doubly-commuting d-tuple of A-hyponormal opera

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14

Müller, Vladimir. "UNIVERSAL N-TUPLES OF OPERATORS." Mathematical Proceedings of the Royal Irish Academy 113A, no. 2 (2013): 143–50. http://dx.doi.org/10.1353/mpr.2013.0015.

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15

Beliakov, G., T. Calvo, and A. Pradera. "Absorbent tuples of aggregation operators." Fuzzy Sets and Systems 158, no. 15 (2007): 1675–91. http://dx.doi.org/10.1016/j.fss.2007.03.007.

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16

Alshammari, Hadi Obaid. "Higher order hyperexpansivity and higher order hypercontractivity." AIMS Mathematics 8, no. 11 (2023): 27227–40. http://dx.doi.org/10.3934/math.20231393.

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<abstract><p>As a natural extension of the concept of $ (m, p) $-hyperexpansive and $ (m, p) $-hypercontractive of a single operator, we introduce and study the concepts of $ (m, p) $-hyperexpansivity and $ (m, p) $-hypercontractivity for $ d $-tuple of commuting operators acting on Banach spaces. These concepts extend the definitions of $ m $-isometries and $ (m, p) $-isometric tuples of bounded linear operators acting on Hilbert or Banach spaces, which have been introduced and studied by many authors.</p></abstract>

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17

Müller, V., and Yu Tomilov. "Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov." Concrete Operators 7, no. 1 (2020): 133–54. http://dx.doi.org/10.1515/conop-2020-0102.

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AbstractWe present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vecto

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18

Amor, Ali Ben. "An extension of Henrici theorem for the joint approximate spectrum of commuting spectral operators." Journal of the Australian Mathematical Society 75, no. 2 (2003): 233–46. http://dx.doi.org/10.1017/s1446788700003748.

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AbstractGiven two m-tuples of commuting spectral operators on a Hilbert space, T = (T1,…, Tm) and S = (S1,…, Sm), an extended version of Henrici perturbation theorem is obtained for the joint approximate spectrum of S under perturbation by T. We also derive an extended version of Bauer-Fike theorem for such tuples of operators. The method used involves Clifford algebra techniques introduced by McIntosh and Pryde.

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19

Rudol, K. "The spectrum of orthogonal sums of subnormal pairs." Glasgow Mathematical Journal 30, no. 1 (1988): 11–15. http://dx.doi.org/10.1017/s0017089500006984.

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This note provides yet another example of the difficulties that arise when one wants to extend the spectral theory of subnormal operators to subnormal tuples. Several basic properties of a subnormal operator Y remain true for tuples; e.g. the existence and uniqueness of its minimal normal extension N, the spectral inclusion σ(N)⊂ σ(Y)-proved for n-tuples in [4] and generalized to infinite tuples in [5]. However, neither the invariant subspace theorem nor the spectral mapping theorem in the “strong form” as in [3] is known so far for subnormal tuples.

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20

Yakubovich, Dmitry, and Sameer Chavan. "Spherical tuples of Hilbert space operators." Indiana University Mathematics Journal 64, no. 2 (2015): 577–612. http://dx.doi.org/10.1512/iumj.2015.64.5471.

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21

Bhattacharyya, Tirthankar. "On tuples of commuting compact operators." Publications of the Research Institute for Mathematical Sciences 32, no. 5 (1996): 785–95. http://dx.doi.org/10.2977/prims/1195162382.

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22

Soltani, R., K. Hedayatian, and B. Khani Robati. "On Supercyclicity of Tuples of Operators." Bulletin of the Malaysian Mathematical Sciences Society 38, no. 4 (2014): 1507–16. http://dx.doi.org/10.1007/s40840-014-0083-z.

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23

Ahmed Ould Ahmed Mahmoud, Sid, and Asma Al Rwaily. "COMMUTING TUPLES OF (n, m)-POWER NORMAL OPERATORS IN HILBERT SPACES." Journal of Mathematical Analysis 15, no. 2 (2024): 1–13. https://doi.org/10.54379/jma-2024-2-1.

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The purpose of this study is to extend the notion of (n, m)-power normal bounded operator of Hilbert space to a finite set of commuting such operators. Similar studies for other classes of operators exist in the literature. The purpose of this paper is to introduce and study the structure of certain special types of several variable operators on a Hilbert space named joint (n, m)-power normal multioperators. This is a generalization of the classes of joint normal and joint n-normal multioperators. We consider a multivariable generalization of these single variable n-normal and (n, m)-normal op

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24

Conway, John B., and Jim Gleason. "Absolute Equivalence and Dirac Operators of Commuting Tuples of Operators." Integral Equations and Operator Theory 51, no. 1 (2005): 57–71. http://dx.doi.org/10.1007/s00020-003-1255-7.

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25

Hoffmann, Philipp. "On (m, p)-isometric operators and operator tuples on normed spaces." Irish Mathematical Society Bulletin 0072 (2013): 31–32. http://dx.doi.org/10.33232/bims.0072.31.32.

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26

Altwaijry, Najla, Silvestru Sever Dragomir, and Kais Feki. "Inequalities and Reverse Inequalities for the Joint A-Numerical Radius of Operators." Axioms 12, no. 3 (2023): 316. http://dx.doi.org/10.3390/axioms12030316.

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In this paper, we aim to establish several estimates concerning the generalized Euclidean operator radius of d-tuples of A-bounded linear operators acting on a complex Hilbert space H, which leads to the special case of the well-known A-numerical radius for d=1. Here, A is a positive operator on H. Some inequalities related to the Euclidean operator A-seminorm of d-tuples of A-bounded operators are proved. In addition, under appropriate conditions, several reverse bounds for the A-numerical radius in single and multivariable settings are also stated.

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27

Kerchy, Laszlo. "Quasianalytic n-tuples of Hilbert space operators." Journal of Operator Theory 81, no. 1 (2018): 3–20. http://dx.doi.org/10.7900/jot.2017sep07.2205.

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The residual and ∗-residual parts of the unitary dilation proved to be especially useful in the study of contractions. A more direct approach to these components, originated in B. Sz.-Nagy, \textit{Acta Sci. Math. (Szeged)} \textbf{11}(1947), 152--157, leads to the concept of unitary asymptote, and opens the way for generalizations to more general settings. In this paper a systematic study of unitary asymptotes of commuting n-tuples of general Hilbert space operators is initiated. Special emphasis is put on the study of the quasianalyticity property, which constitutes homogeneous behaviour in

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28

Guterman, A. E., and P. M. Shteyner. "Linear Operators Preserving Majorization of Matrix Tuples." Vestnik St. Petersburg University, Mathematics 53, no. 2 (2020): 136–44. http://dx.doi.org/10.1134/s1063454120020077.

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29

Guterman, Alexander E., and Pavel M. Shteyner. "Linear operators preserving majorization of matrix tuples." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 7 (65), no. 2 (2020): 217–29. http://dx.doi.org/10.21638/11701/spbu01.2020.204.

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30

Fedotov, S. N. "Framed moduli spaces and tuples of operators." Journal of Mathematical Sciences 193, no. 4 (2013): 606–21. http://dx.doi.org/10.1007/s10958-013-1488-1.

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31

Shkarin, Stanislav. "Hypercyclic tuples of operators on ℂnand ℝn". Linear and Multilinear Algebra 60, № 8 (2012): 885–96. http://dx.doi.org/10.1080/03081087.2010.533174.

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32

Müller, Vladimir. "Universal N-Tuples of Operators." Mathematical Proceedings of the Royal Irish Academy 113, no. 2 (2013): 143–50. http://dx.doi.org/10.3318/pria.2013.113.13.

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33

Bhatia, Rajendra, Ludwig Elsner, and Peter ?emrl. "Distance between commuting tuples of normal operators." Archiv der Mathematik 71, no. 3 (1998): 229–32. http://dx.doi.org/10.1007/s000130050257.

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34

Jeon, In Ho. "On joint essential spectra of doubly commuting n-tuples of p-hyponormal operators." Glasgow Mathematical Journal 40, no. 3 (1998): 353–58. http://dx.doi.org/10.1017/s0017089500032705.

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AbstractLet A be an operator on a Hillbert space with polar decomposition A = |A|, let Â = |A|½U|A|½ and let Â = V|Â| be the polar decomposition of Â. Write Ã for the operatorÃ = |Â|½V|Â|½. If = (A1,…,AN) is a doubly commuting n-tuple of p-hyponormal operators on a Hillbert space with equal defect and nullity, then = (Ã1,…,Ãn) is a doubly commuting n-tuple of hyponormal operators. In this paper we show thatwhere σ* denotes σTe (Taylor essential spectrum), σTw (Taylor-Weyl spectrum) and σTb (Taylor-Browder spectrum), respectively.

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35

Hladnik, Milan. "Spectrality of elementary operators." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 49, no. 2 (1990): 327–46. http://dx.doi.org/10.1017/s1446788700030603.

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AbstractSpectrality and prespectrality of elementary operators , acting on the algebra B(k) of all bounded linear operators on a separable infinite-dimensional complex Hubert space K, or on von Neumann-Schatten classes in B(k), are treated. In the case when (a1, a2, …, an) and (b1, b2, …, bn) are two n—tuples of commuting normal operators on H, the complete characterization of spectrality is given.

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36

Ahmed Mahmoud, Sid Ahmed Ould, Ahmed Himadan Ahmed та Ahmad Sarosh. "α , β -Normal Operators in Several Variables". Mathematical Problems in Engineering 2022 (15 червня 2022): 1–11. http://dx.doi.org/10.1155/2022/3020449.

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We consider an extension of the concept of α , β -normal operators in single variable operator to tuples of operators, similar to those extensions of the concepts of normality to joint normality, hyponormality to joint hyponormality, and quasi-hyponormality to joint quasi-hyponormality.

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37

Ahmad, Naeem, E. S. Kamel, and Sid Mahmoud. "(N1,..., nd)-quasi-(p, q)-isometric commuting tuple of operators." Filomat 37, no. 28 (2023): 9531–42. http://dx.doi.org/10.2298/fil2328531a.

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In this work we construct the concept based on the extension of n-quasi-p-isometric operators of a single operator studied in [11, 14] to the multi-dimentional operators. we are introducing some new interesting results of these family of tuples of operators that are expanding some results of recently published works based on a single operator.

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38

Friedrich, J. "On tuples of commuting symmetric, non-selfadjoint operators." Integral Equations and Operator Theory 13, no. 4 (1990): 553–75. http://dx.doi.org/10.1007/bf01210401.

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39

Feldman, Nathan S. "Hypercyclic tuples of operators and somewhere dense orbits." Journal of Mathematical Analysis and Applications 346, no. 1 (2008): 82–98. http://dx.doi.org/10.1016/j.jmaa.2008.04.027.

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40

Nazarov, Fedor, and Vladimir Peller. "Functions of perturbed tuples of self-adjoint operators." Comptes Rendus Mathematique 350, no. 7-8 (2012): 349–54. http://dx.doi.org/10.1016/j.crma.2012.04.010.

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41

Xia, Daoxing. "A Commutator Formula for Subnormal Tuples of Operators." Integral Equations and Operator Theory 83, no. 3 (2015): 381–92. http://dx.doi.org/10.1007/s00020-015-2238-1.

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42

Naeem, Ahmad. "On m-expansive and m-contractives tuple operators in Hilbert spaces." Annals of Communications in Mathematics 3, no. 3 (2020): 199–207. https://doi.org/10.5281/zenodo.10048547.

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43

Duggal, B. P. "On the spectrum of n-tuples of p-hyponormal operators." Glasgow Mathematical Journal 40, no. 1 (1998): 123–31. http://dx.doi.org/10.1017/s0017089500032419.

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Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let ℋU(p) denot

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44

Qiyas, Muhammad, Muhammad Naeem, Lazim Abdullah, Muhammad Riaz, and Neelam Khan. "Decision Support System Based on Complex Fractional Orthotriple Fuzzy 2-Tuple Linguistic Aggregation Operator." Symmetry 15, no. 1 (2023): 251. http://dx.doi.org/10.3390/sym15010251.

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In this research, we provide tools to overcome the information loss limitation resulting from the requirement to estimate the results in the discrete initial expression domain. Through the use of 2-tuples, which are made up of a linguistic term and a numerical value calculated between [0.5,0.5), the linguistic information will be expressed. This model supports continuous representation of the linguistic data within its scope, permitting it to express any information counting received through an aggregation procedure. This study provides a novel approach to develop a linguistic multi-attribute

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45

Benczúr, András, and Gyula I. Szabó. "Towards a Normal Form and a Query Language for Extended Relations Defined by Regular Expressions." Journal of Database Management 27, no. 2 (2016): 27–48. http://dx.doi.org/10.4018/jdm.2016040102.

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This paper introduces a generalized data base concept that unites relational and semi structured data models. As an important theoretical result we could find a quadratic decision algorithm for the implication problem of functional and join dependencies defined on the united data model. As practical contribution we presented a normal form for the new data model as a tool for data base design. With our novel representations of regular expressions, a more effective searching method could be developed. XML elements are described by XML schema languages such as a DTD or an XML Schema definition. T

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46

Xia, Daoxing. "On pure subnormal operators with finite rank self-commutators and related operator tuples." Integral Equations and Operator Theory 24, no. 1 (1996): 106–25. http://dx.doi.org/10.1007/bf01195487.

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47

Cobos, Fernando, Luz M. Fernández-Cabrera, and Joaquim Martín. "Some reiteration results for interpolation methods defined by means of polygons." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 138, no. 6 (2008): 1179–95. http://dx.doi.org/10.1017/s0308210507000315.

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We continue the research on reiteration results between interpolation methods associated to polygons and the real method. Applications are given to N-tuples of function spaces, of spaces of bounded linear operators and Banach algebras.

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48

Gerhold, Malte, and Orr Moshe Shalit. "On the matrix range of random matrices." Journal of Operator Theory 85, no. 2 (2021): 527–45. http://dx.doi.org/10.7900/jot.2019dec04.2277.

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This note treats a simple minded question: \textit{what does a typical random matrix range look like?} We study the relationship between various modes of convergence for tuples of operators on the one hand, and continuity of matrix ranges with respect to the Hausdorff metric on the other. In particular, we show that the matrix range of a tuple generating a continuous field of C∗-algebras is continuous in the sense that every level is continuous in the Hausdorff metric. Using this observation together with known results on strong convergence in distribution of matrix ensembles, we identify the

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Andrei, ِDiana. "Multicentric holomorphic calculus for $n-$tuples of commuting operators." Advances in Operator Theory 4, no. 2 (2019): 447–61. http://dx.doi.org/10.15352/aot.1804-1346.

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Giménez, José. "Joint spectrum of subnormal $n$-tuples of composition operators." Proceedings of the American Mathematical Society 130, no. 7 (2001): 2015–23. http://dx.doi.org/10.1090/s0002-9939-01-06304-3.

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