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Journal articles on the topic 'Invariant subspaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 30 July 2024

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1

IZUCHI, KEI JI, KOU HEI IZUCHI, and YUKO IZUCHI. "SPLITTING INVARIANT SUBSPACES IN THE HARDY SPACE OVER THE BIDISK." Journal of the Australian Mathematical Society 102, no. 2 (May 12, 2016): 205–23. http://dx.doi.org/10.1017/s1446788716000203.

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Let $H^{2}$ be the Hardy space over the bidisk. It is known that Hilbert–Schmidt invariant subspaces of $H^{2}$ have nice properties. An invariant subspace which is unitarily equivalent to some invariant subspace whose continuous spectrum does not coincide with $\overline{\mathbb{D}}$ is Hilbert–Schmidt. We shall introduce the concept of splittingness for invariant subspaces and prove that they are Hilbert–Schmidt.

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2

Wang, Guo-Hua, Jia-Fu Pang, Yong-Yang Jin, and Bo Ren. "Invariant Subspaces of Short Pulse-Type Equations and Reductions." Symmetry 16, no. 6 (June 18, 2024): 760. http://dx.doi.org/10.3390/sym16060760.

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In this paper, we extend the invariant subspace method to a class of short pulse-type equations. Complete classification results with invariant subspaces from 2 to 5 dimensions are provided. The key step is to take subspaces of solutions of linear ordinary differential equations as invariant subspaces that nonlinear operators admit. Some concrete examples and corresponding reduced systems are presented to illustrate this method.

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3

ASSADI, AMANOLLAH, MOHAMAD ALI FARZANEH, and HAJI MOHAMMAD MOHAMMADINEJAD. "ON THE DECOMPOSITION OF OPERATORS WITH SEVERAL ALMOST-INVARIANT SUBSPACES." Bulletin of the Australian Mathematical Society 99, no. 2 (January 4, 2019): 274–83. http://dx.doi.org/10.1017/s0004972718001363.

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We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

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4

Bose, Snehasish, P. Muthukumar, and Jaydeb Sarkar. "Beurling type invariant subspaces of composition operators." Journal of Operator Theory 86, no. 2 (November 15, 2021): 425–38. http://dx.doi.org/10.7900/jot.2020may15.2286.

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The aim of this paper is to answer the following question concerning invariant subspaces of composition operators: characterize φ, holomorphic self maps of D, and inner functions θ∈H∞(D) such that the Beurling type invariant subspace θH2 is an invariant subspace for Cφ. We prove the following result: Cφ(θH2)⊆θH2 if and only if θ∘φθ∈S(D). This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.

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5

Liu, Junfeng. "On Invariant Subspaces for the Shift Operator." Symmetry 11, no. 6 (June 1, 2019): 743. http://dx.doi.org/10.3390/sym11060743.

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In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace M in the Hardy space H p ( D ) ( 1 ≤ p < ∞ ) is invariant under the shift operator M z on H p ( D ) if and only if it is hyperinvariant under M z , and that a closed linear subspace M in the Lebesgue space L 2 ( ∂ D ) is reducing under the shift operator M e i θ on L 2 ( ∂ D ) if and only if it is hyperinvariant under M e i θ . At the same time, we show that there are two large classes of invariant subspaces for M e i θ that are not hyperinvariant subspaces for M e i θ and are also not reducing subspaces for M e i θ . Moreover, we still show that there is a large class of hyperinvariant subspaces for M z that are not reducing subspaces for M z . Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space H 2 ( D ) , which are the analogue of the formula of the reproducing function in the Bergman space A 2 ( D ) . In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.

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6

Krivosheev, A. S., and O. A. Krivosheeva. "Invariant Subspaces in Unbounded Domains." Issues of Analysis 28, no. 3 (November 2021): 91–107. http://dx.doi.org/10.15393/j3.art.2021.10870.

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7

Szekelyhidi, Laszlo, and Seyyed Mohammad Tabatabaie. "Invariant Subspaces on KPC-Hypergroups." Zurnal matematiceskoj fiziki, analiza, geometrii 15, no. 1 (March 25, 2019): 122–30. http://dx.doi.org/10.15407/mag15.01.122.

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8

Forouzanfar, AM, S. Khorshidvandpour, and Z. Bahmani. "Uniformly invariant normed spaces." BIBECHANA 10 (October 31, 2013): 31–33. http://dx.doi.org/10.3126/bibechana.v10i0.7555.

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In this work, we introduce the concepts of compactly invariant and uniformly invariant. Also we define sometimes C-invariant closed subspaces and then prove every m-dimensional normed space with m > 1 has a nontrivial sometimes C-invariant closed subspace. Sequentially C-invariant closed subspaces are also introduced. Next, An open problem on the connection between compactly invariant and uniformly invariant normed spaces has been posed. Finally, we prove a theorem on the existence of a positive operator on a strict uniformly invariant Hilbert space. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.7555 BIBECHANA 10 (2014) 31-33

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9

Fernández-Morales, H. R., A. G. García, M. A. Hernández-Medina, and M. J. Muñoz-Bouzo. "Generalized sampling: From shift-invariant to U-invariant spaces." Analysis and Applications 13, no. 03 (March 5, 2015): 303–29. http://dx.doi.org/10.1142/s0219530514500213.

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The aim of this article is to derive a sampling theory in U-invariant subspaces of a separable Hilbert space ℋ where U denotes a unitary operator defined on ℋ. To this end, we use some special dual frames for L2(0, 1), and the fact that any U-invariant subspace with stable generator is the image of L2(0, 1) by means of a bounded invertible operator. The used mathematical technique mimics some previous sampling work for shift-invariant subspaces of L2(ℝ). Thus, sampling frame expansions in U-invariant spaces are obtained. In order to generalize convolution systems and deal with the time-jitter error in this new setting we consider a continuous group of unitary operators which includes the operator U.

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10

Guo, Kunyu, and Dechao Zheng. "Invariant Subspaces, Quasi-invariant Subspaces, and Hankel Operators." Journal of Functional Analysis 187, no. 2 (December 2001): 308–42. http://dx.doi.org/10.1006/jfan.2001.3820.

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11

AHMADI, A., A. ASKARI HEMMAT, and R. RAISI TOUSI. "SHIFT INVARIANT SPACES FOR LOCAL FIELDS." International Journal of Wavelets, Multiresolution and Information Processing 09, no. 03 (May 2011): 417–26. http://dx.doi.org/10.1142/s0219691311004122.

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This paper is an investigation of shift invariant subspaces of L2(G), where G is a locally compact abelian group, or in general a local field, with a compact open subgroup. In this paper we state necessary and sufficient conditions for shifts of an element of L2(G) to be an orthonormal system or a Parseval frame. Also we show that each shift invariant subspace of L2(G) is a direct sum of principle shift invariant subspaces of L2(G) generated by Parseval frame generators.

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12

Eschmeier, Jörg. "Wandering subspace property for homogeneous invariant subspaces." Banach Journal of Mathematical Analysis 13, no. 2 (April 2019): 486–505. http://dx.doi.org/10.1215/17358787-2018-0052.

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13

Nakazi, Takahiko. "Multipliers of invariant subspaces in the bidisc." Proceedings of the Edinburgh Mathematical Society 37, no. 2 (June 1994): 193–99. http://dx.doi.org/10.1017/s0013091500006003.

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For any nonzero invariant subspace M in H2(T2), set . Then Mx is also an invariant subspace of H2(T2) that contains M. If M is of finite codimension in H2(T2) then Mx = H2(T2) and if M = qH2(T2) for some inner function q then Mx = M. In this paper invariant subspaces with Mx = M are studied. If M = q1H2(T2) ∩ q2H2(T2) and q1, q2 are inner functions then Mx = M. However in general this invariant subspace may not be of the form: qH2(T2) for some inner function q. Put (M) = {ø ∈ L ∞: ø M ⊆ H2(T2)}; then (M) is described and (M) = (Mx) is shown. This is the set of all multipliers of M in the title. A necessary and sufficient condition for (M) = H∞(T2) is given. It is noted that the kernel of a Hankel operator is an invariant subspace M with Mx = M. The argument applies to the polydisc case.

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14

Mecheri, Salah. "Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators." Georgian Mathematical Journal 29, no. 2 (November 30, 2021): 233–44. http://dx.doi.org/10.1515/gmj-2021-2124.

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Abstract The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

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15

Lauric, Vasile. "Some remarks on the invariant subspace problem for hyponormal operators." International Journal of Mathematics and Mathematical Sciences 28, no. 6 (2001): 359–65. http://dx.doi.org/10.1155/s0161171201011966.

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We make some remarks concerning the invariant subspace problem for hyponormal operators. In particular, we bring together various hypotheses that must hold for a hyponormal operator without nontrivial invariant subspaces, and we discuss the existence of such operators.

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16

Ilic, M., and Ian W. Turner. "Approximately invariant subspaces." ANZIAM Journal 44 (April 1, 2003): 378. http://dx.doi.org/10.21914/anziamj.v44i0.687.

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17

Andersson, Lars. "Invariant Lagrangian subspaces." Proceedings of the American Mathematical Society 103, no. 4 (April 1, 1988): 1113. http://dx.doi.org/10.1090/s0002-9939-1988-0954992-7.

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18

Gichev, V. M., and S. M. Dobrovol'skii. "Extremal invariant subspaces." Siberian Mathematical Journal 25, no. 6 (1985): 860–72. http://dx.doi.org/10.1007/bf00968940.

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19

Maji, Amit, Aneesh Mundayadan, Jaydeb Sarkar, and T. R. Sankar. "Characterization of invariant subspaces in the polydisc." Journal of Operator Theory 82, no. 2 (September 15, 2019): 445–68. http://dx.doi.org/10.7900/jot.2018jun07.2204.

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We give a complete characterization of invariant subspaces for (Mz1,…,Mzn) on the Hardy space H2(Dn) over the unit polydisc Dn in Cn, n>1. In particular, this yields a complete set of unitary invariants for invariant subspaces for (Mz1,…,Mzn) on H2(Dn). As a consequence, we classify a large class of n-tuples of commuting isometries. All of our results hold for vector-valued Hardy spaces over Dn, n>1.

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20

Carmo, Joao R., and S. Waleed Noor. "Universal composition operators." Journal of Operator Theory 87, no. 1 (December 15, 2021): 137–56. http://dx.doi.org/10.7900/jot.2020aug03.2301.

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A Hilbert space operator U is called \textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of U restricted to one of its invariant subspaces. It follows that the \textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for U are one dimensional. In this article we characterize all linear fractional composition operators Cϕf=f∘ϕ that have universal translates on both the classical Hardy spaces H2(C+) and H2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H2(D) with affine symbol ϕa(z)=az+(1−a) for $0

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21

Hare, K. E., and J. A. Ward. "Finite dimensional H-invariant spaces." Bulletin of the Australian Mathematical Society 56, no. 3 (December 1997): 353–61. http://dx.doi.org/10.1017/s0004972700031142.

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A subset V of M(G) is left H-invariant if it is invariant under left translation by the elements of H, a subset of a locally compact group G. We establish necessary and sufficient conditions on H which ensure that finite dimensional subspaces of M(G) when G is compact, or of L∞(G) when G is locally compact Abelian, which are invariant in this weaker sense, contain only trigonometric polynomials. This generalises known results for finite dimensional G-invariant subspaces. We show that if H is a subgroup of finite index in a compact group G, and the span of the H-translates of μ is a weak*-closed subspace of L∞(G) or M(G) (or is closed in Lp(G)for 1 ≤ p < ∞), then μ is a trigonometric polynomial.We also obtain some results concerning functions that possess the analogous weaker almost periodic condition relative to H.

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22

Izuchi, Keiji, and Yasuo Matsugu. "Aϕ-Invariant Subspaces on the Torus." Canadian Journal of Mathematics 50, no. 1 (February 1, 1998): 99–133. http://dx.doi.org/10.4153/cjm-1998-006-0.

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AbstractGeneralizing the notion of invariant subspaces on the 2-dimensional torus T2, we study the structure of Aϕ-invariant subspaces of L2(T2). A complete description is given of Aϕ-invariant subspaces that satisfy conditions similar to those studied by Mandrekar, Nakazi, and Takahashi.

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23

Young, Robert M. "74.43 On Invariant Subspaces." Mathematical Gazette 74, no. 470 (December 1990): 368. http://dx.doi.org/10.2307/3618137.

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24

Petersson, Henrik. "HYPERCYCLICITY ON INVARIANT SUBSPACES." Journal of the Korean Mathematical Society 45, no. 4 (July 31, 2008): 903–21. http://dx.doi.org/10.4134/jkms.2008.45.4.903.

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25

Bercovici, Hari. "Notes on invariant subspaces." Bulletin of the American Mathematical Society 23, no. 1 (July 1, 1990): 1–37. http://dx.doi.org/10.1090/s0273-0979-1990-15894-x.

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26

Lang, Patrick, Ann Gironella, and Rieken Venema. "Invariant Subspaces and Regression." Communications in Statistics - Theory and Methods 42, no. 3 (February 2013): 491–504. http://dx.doi.org/10.1080/03610926.2011.581784.

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27

Kissin, E. V. "Invariant subspaces for derivations." Proceedings of the American Mathematical Society 102, no. 1 (January 1, 1988): 95. http://dx.doi.org/10.1090/s0002-9939-1988-0915723-x.

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28

Noakes, Lyle. "Invariant subspaces and perturbations." Proceedings of the American Mathematical Society 114, no. 2 (February 1, 1992): 365. http://dx.doi.org/10.1090/s0002-9939-1992-1087467-9.

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29

Apostol, Constantin, Ciprian Foias, and Norberto Salinas. "On stable invariant subspaces." Integral Equations and Operator Theory 8, no. 6 (November 1985): 721–50. http://dx.doi.org/10.1007/bf01213789.

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30

Kovarik, Zdislav V., and Nagwa Sherif. "Perturbation of invariant subspaces." Linear Algebra and its Applications 64 (January 1985): 93–113. http://dx.doi.org/10.1016/0024-3795(85)90269-1.

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31

Xu, Xiao-Ming, Xiaochun Fang, and Fugen Gao. "Principal invariant subspaces theorems." Linear and Multilinear Algebra 62, no. 11 (September 6, 2013): 1428–36. http://dx.doi.org/10.1080/03081087.2013.832245.

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32

Baranov, Anton, and Yurii Belov. "Synthesizable differentiation-invariant subspaces." Geometric and Functional Analysis 29, no. 1 (February 2019): 44–71. http://dx.doi.org/10.1007/s00039-019-00474-8.

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33

Prakash, R. "EXISTENCE OF INVARIANT SUBSPACES." International Journal of Functional Analysis, Operator Theory and Applications 10, no. 2 (September 12, 2018): 73–80. http://dx.doi.org/10.17654/fa010020073.

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34

Djordjević, Slaviša V., Robin E. Harte, and David R. Larson. "Partially hyper invariant subspaces." Operators and Matrices, no. 1 (2012): 97–106. http://dx.doi.org/10.7153/oam-06-07.

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35

Mello, A., and C. S. Kubrusly. "Quasiaffinity and invariant subspaces." Archiv der Mathematik 107, no. 2 (June 16, 2016): 173–84. http://dx.doi.org/10.1007/s00013-016-0919-x.

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36

Xu, Xiao Ming, and Xiao Chun Fang. "On principal invariant subspaces." Acta Mathematica Sinica, English Series 31, no. 10 (September 15, 2015): 1621–28. http://dx.doi.org/10.1007/s10114-015-4362-x.

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37

Makarov, Konstantin A., Stephan Schmitz, and Albrecht Seelmann. "On Invariant Graph Subspaces." Integral Equations and Operator Theory 85, no. 3 (May 5, 2016): 399–425. http://dx.doi.org/10.1007/s00020-016-2297-y.

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38

Omladič, Matjaž. "Perturbation of Invariant Subspaces." Journal of Mathematical Analysis and Applications 197, no. 1 (January 1996): 125–37. http://dx.doi.org/10.1006/jmaa.1996.0011.

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39

Dieci, L., and M. J. Friedman. "Continuation of invariant subspaces." Numerical Linear Algebra with Applications 8, no. 5 (2001): 317–27. http://dx.doi.org/10.1002/nla.245.

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40

Bergqvist, Linus. "Alternative proofs of Mandrekar’s theorem." Proceedings of the American Mathematical Society, Series B 10, no. 4 (February 27, 2023): 46–55. http://dx.doi.org/10.1090/bproc/156.

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We present two alternative proofs of Mandrekar’s theorem, which states that an invariant subspace of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition [Proc. Amer. Math. Soc. 103 (1988), pp. 145–148]. The first proof uses properties of Toeplitz operators to derive a formula for the reproducing kernel of certain shift invariant subspaces, which can then be used to characterize them. The second proof relies on the reproducing property in order to show that the reproducing kernel at the origin must generate the entire shift invariant subspace.

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41

Galaktionov, Victor A. "On invariant subspaces for nonlinear finite-difference operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 6 (1998): 1293–308. http://dx.doi.org/10.1017/s0308210500027335.

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We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.

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42

Aleman, Alexandru, Stefan Richter, and William T. Ross. "Bergman Spaces on Disconnected Domains." Canadian Journal of Mathematics 48, no. 2 (April 1, 1996): 225–43. http://dx.doi.org/10.4153/cjm-1996-011-5.

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AbstractFor a bounded region G ⊂ ℂ and a compact set K ⊂ G, with area measure zero, we will characterize the invariant subspaces ℳ (under ƒ → zƒ) of the Bergman space (G \ K), 1 ≤ p < ∞, which contain (G) and with dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K. When G \ K is connected, we will see that dim(ℳ/(z - λ)ℳ) = 1 for all λ ∈ G \ K and thus in this case we will have a complete description of the invariant subspaces lying between (G) and (G \ K). When p = ∞, we will remark on the structure of the weak-star closed z-invariant subspaces between H∞(G) and H∞(G \ K). When G \ K is not connected, we will show that in general the invariant subspaces between (G) and (G \ K) are fantastically complicated. As an application of these results, we will remark on the complexity of the invariant subspaces (under ƒ → ζƒ) of certain Besov spaces on K. In particular, we shall see that in the harmonic Dirichlet space , there are invariant subspaces ℱ such that the dimension of ζℱ in ℱ is infinite.

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43

Beattie, Christopher, Mark Embree, and John Rossi. "Convergence of Restarted Krylov Subspaces to Invariant Subspaces." SIAM Journal on Matrix Analysis and Applications 25, no. 4 (January 2004): 1074–109. http://dx.doi.org/10.1137/s0895479801398608.

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44

Hentosh, O. Ye. "The Bargmann type reduction for some Lax integrable two-dimensional generalization of the relativistic Toda lattice." Carpathian Mathematical Publications 7, no. 2 (December 19, 2015): 155–71. http://dx.doi.org/10.15330/cmp.7.2.155-171.

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The possibility of applying the method of reducing upon finite-dimensional invariant subspaces, generated by the eigenvalues of the associated spectral problem, to some two-dimensional generalization of the relativistic Toda lattice with the triple matrix Lax type linearization is investigated. The Hamiltonian property and Lax-Liouville integrability of the vector fields, given by this system, on the invariant subspace related with the Bargmann type reduction are found out.

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45

Song, Junquan, Yujian Ye, Danda Zhang, and Jun Zhang. "Conditional Lie-Bäcklund Symmetries and Reductions of the Nonlinear Diffusion Equations with Source." Abstract and Applied Analysis 2014 (2014): 1–17. http://dx.doi.org/10.1155/2014/898032.

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Conditional Lie-Bäcklund symmetry approach is used to study the invariant subspace of the nonlinear diffusion equations with sourceut=e−qx(epxP(u)uxm)x+Q(x,u),m≠1. We obtain a complete list of canonical forms for such equations admit multidimensional invariant subspaces determined by higher order conditional Lie-Bäcklund symmetries. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamic systems.

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46

Fuller, K. R., W. K. Nicholson, and J. F. Watters. "Reflexive Bimodules." Canadian Journal of Mathematics 41, no. 4 (August 1, 1989): 592–611. http://dx.doi.org/10.4153/cjm-1989-026-x.

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If VK is a finite dimensional vector space over a field K and L is a lattice of subspaces of V, then, following Halmos [11], alg L is defined to be (the K-algebra of) all K-endomorphisms of V which leave every subspace in L invariant. If R ⊆ end(VK) is any subalgebra we define lat R to be (the sublattice of) all subspaces of VK which are invariant under every transformation in R. Then R ⊆ alg [lat R] and R is called a reflexive algebra when this is equality. Every finite dimensional algebra is isomorphic to a reflexive one ([4]) and these reflexive algebras have been studied by Azoff [1], Barker and Conklin [3] and Habibi and Gustafson [9] among others.

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47

Raynaud, Yves. "A note on symmetric basic sequences in Lp(Lq)." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 1 (July 1992): 183–94. http://dx.doi.org/10.1017/s0305004100070869.

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Subspaces of Lp spanned by symmetric independent identically distributed random variables were identified as Orlicz spaces by Bretagnolle and Dacunha-Castelle[1], who showed that, conversely, in the case p ≤ 2, every p-convex, 2-concave Orlicz space is isomorphic to a subspace of Lp. This was extended by Dacunha-Castelle [3] to subspaces of Lp with symmetric basis, which appear as ‘p-means’ of Orlicz spaces (see [9] for the corresponding finite-dimensional result, and [12] for the case of rearrangement invariant function spaces). On the contrary the only subspaces with symmetric basis of Lp for p ≥ 2 are lp and l2 (if one does not care about isomorphy constants).

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48

Zhao, Junjian, Wei-Shih Du, and Yasong Chen. "New Generalizations and Results in Shift-Invariant Subspaces of Mixed-Norm Lebesgue Spaces \({L_{\vec{p}}(\mathbb{R}^d)}\)." Mathematics 9, no. 3 (January 25, 2021): 227. http://dx.doi.org/10.3390/math9030227.

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In this paper, we establish new generalizations and results in shift-invariant subspaces of mixed-norm Lebesgue spaces Lp→(Rd). We obtain a mixed-norm Hölder inequality, a mixed-norm Minkowski inequality, a mixed-norm convolution inequality, a convolution-Hölder type inequality and a stability theorem to mixed-norm case in the setting of shift-invariant subspace of Lp→(Rd). Our new results unify and refine the existing results in the literature.

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49

Chattopadhyay, Arup, Jaydeb Sarkar, and Srijan Sarkar. "Multiplicities, invariant subspaces and an additive formula." Proceedings of the Edinburgh Mathematical Society 64, no. 2 (April 30, 2021): 279–97. http://dx.doi.org/10.1017/s0013091521000146.

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AbstractLet $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$. If $\mathcal{Q}_i^{\bot }$, $i = 1, \ldots , n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$-invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by \[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \]A similar result holds for the Bergman space over the unit polydisc.

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Albertini, Francesca, and Domenico D'Alessandro. "Symmetric states and dynamics of three quantum bits." Quantum Information and Computation 22, no. 7&8 (May 2022): 541–68. http://dx.doi.org/10.26421/qic22.7-8-1.

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The unitary group acting on the Hilbert space ${\cal H}:=(C^2)^{\otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${\cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $\frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.

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