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Journal articles on the topic 'Commutant Lifting Theorem'

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Published: 6 September 2023

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1

Bercovici, Hari, Ciprian Foias, and Allen Tannenbaum. "A spectral commutant lifting theorem." Transactions of the American Mathematical Society 325, no. 2 (February 1, 1991): 741–63. http://dx.doi.org/10.1090/s0002-9947-1991-1000144-9.

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2

Szehr, Oleg, and Rachid Zarouf. "Interpolation without commutants." Journal of Operator Theory 84, no. 1 (May 15, 2020): 239–56. http://dx.doi.org/10.7900/jot.2019may21.2264.

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We introduce a ``dual-space approach'' to mixed Nevanlinna--Pick Carath\'eodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting one of D. Sarason and B. Nagy-C. Foia\c{s}. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This functional extension lemma plays a similar role as Sarason's commutant lifting theorem but it only involves the predual of X and no Hilbert space structure is needed. As an example, we present the respective Pick-type interpolation theorems for Beurling-Sobolev spaces.

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3

Popescu, Gelu. "Andô dilations and inequalities on non-commutative domains." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (July 6, 2018): 1239–67. http://dx.doi.org/10.1017/s030821051800015x.

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We obtain intertwining dilation theorems for non-commutative regular domains 𝒟f and non-commutative varieties 𝒱J in B(𝓗)n, which generalize Sarason and Szőkefalvi-Nagy and Foiaş's commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain 𝒟f (respectively, variety 𝒱J ) as well as a Schur-type representation for the unit ball of the Hardy algebra associated with the variety 𝒱J. We provide Andô-type dilations and inequalities for bi-domains 𝒟f ×c 𝒟g consisting of all pairs (X,Y ) of tuples X := (X1,…, Xn1) ∊ 𝒟f and Y := (Y1,…, Yn2) ∊ 𝒟g that commute, i.e. each entry of X commutes with each entry of Y . The results are new, even when n1 = n2 = 1. In this particular case, we obtain extensions of Andô's results and Agler and McCarthy's inequality for commuting contractions to larger classes of commuting operators. All the results are extended to bi-varieties 𝒱J1×c 𝒱J2, where 𝒱J1 and 𝒱J2 are non-commutative varieties generated by weak-operator-topology-closed two-sided ideals in non-commutative Hardy algebras. The commutative case and the matrix case when n1 = n2 = 1 are also discussed.

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4

Popescu, Gelu. "!COMMUTANT LIFTING, TENSOR ALGEBRAS, AND FUNCTIONAL CALCULUS." Proceedings of the Edinburgh Mathematical Society 44, no. 2 (June 2001): 389–406. http://dx.doi.org/10.1017/s0013091598001059.

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AbstractA non-commutative multivariable analogue of Parrott’s generalization of the Sz.-Nagy–Foia\c{s} commutant lifting theorem is obtained. This yields Tomita-type commutant results and interpolation theorems (e.g. Sarason, Nevanlinna–Pick, Carathéodory) for $F_n^\infty\,\bar{\otimes}\,\M$, the weakly-closed algebra generated by the spatial tensor product of the non-commutative analytic Toeplitz algebra $F_n^\infty$ and an arbitrary von Neumann algebra $\M$. In particular, we obtain interpolation theorems for bounded analytic functions from the open unit ball of $\mathbb{C}^n$ into a von Neumann algebra.A variant of the non-commutative Poisson transform is used to extend the von Neumann inequality to tensor algebras, and to provide a generalization of the functional calculus for contractive sequences of operators on Hilbert spaces. Commutative versions of these results are also considered.AMS 2000 Mathematics subject classification: Primary 47L25; 47A57; 47A60. Secondary 30E05

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5

Ball, J. A., W. S. Li, D. Timotin, and T. T. Trent. "A commutant lifting theorem on the polydisc." Indiana University Mathematics Journal 48, no. 2 (1999): 0. http://dx.doi.org/10.1512/iumj.1999.48.1708.

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6

Sultanic, Saida. "Commutant Lifting Theorem for the Bergman Space." Integral Equations and Operator Theory 55, no. 4 (May 3, 2006): 573–95. http://dx.doi.org/10.1007/s00020-006-1442-4.

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7

Gu, Caixing. "On causality in commutant lifting theorem. I." Integral Equations and Operator Theory 16, no. 1 (March 1993): 82–97. http://dx.doi.org/10.1007/bf01196603.

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8

Timotin, Dan. "Completions of matrices and the commutant lifting theorem." Journal of Functional Analysis 104, no. 2 (March 1992): 291–98. http://dx.doi.org/10.1016/0022-1236(92)90002-z.

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9

Freydin, Boris. "Commutant lifting theorem and interpolation in discrete nest algebras." Integral Equations and Operator Theory 29, no. 2 (June 1997): 211–30. http://dx.doi.org/10.1007/bf01191431.

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10

Timotin, Dan. "The weighted commutant lifting theorem in the coupling approach." Integral Equations and Operator Theory 42, no. 4 (December 2002): 493–97. http://dx.doi.org/10.1007/bf01270926.

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11

Ball, Joseph A., and Israel Gohberg. "A commutant lifting theorem for triangular matrices with diverse applications." Integral Equations and Operator Theory 8, no. 2 (March 1985): 205–67. http://dx.doi.org/10.1007/bf01202814.

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12

Agler, J., and N. J. Young. "A Commutant Lifting Theorem for a Domain in C2and Spectral Interpolation." Journal of Functional Analysis 161, no. 2 (February 1999): 452–77. http://dx.doi.org/10.1006/jfan.1998.3362.

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13

Frazho, Arthur E. "A note on the commutant lifting theorem and a Generalized four block problem." Integral Equations and Operator Theory 14, no. 2 (March 1991): 299–303. http://dx.doi.org/10.1007/bf01199910.

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14

Foias, C., A. E. Frazho, I. Gohberg, and M. A. Kaashoek. "A time-variant version of the commutant lifting theorem and nonstationary interpolation problems." Integral Equations and Operator Theory 28, no. 2 (June 1997): 158–90. http://dx.doi.org/10.1007/bf01191816.

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15

Foias, Ciprian, Caixing Gu, and Allen Tannenbaum. "Nonlinearity inH ?-control theory, causality in the commutant lifting theorem, and extension of intertwining operators." Integral Equations and Operator Theory 23, no. 1 (March 1995): 89–100. http://dx.doi.org/10.1007/bf01261204.

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16

Marcantognini, S. A. M., and M. D. Morán. "A Schur Analysis of the Minimal Weak Unitary Dilations of a Contraction Operator and the Relaxed Commutant Lifting Theorem." Integral Equations and Operator Theory 64, no. 2 (June 2009): 273–99. http://dx.doi.org/10.1007/s00020-009-1686-x.

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17

Bruzual, Ramon, and Marisela Dominguez. "Equivalence between the dilation and lifting properties of an ordered group through multiplicative families of isometries. A version of the commutant lifting theorem on some lexicographic groups." Integral Equations and Operator Theory 40, no. 1 (March 2001): 1–15. http://dx.doi.org/10.1007/bf01202951.

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18

Burdak, Zbigniew, and Wiesław Grygierzec. "On dilation and commuting liftings of n-tuples of commuting Hilbert space contractions." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 19, no. 1 (December 1, 2020): 121–39. http://dx.doi.org/10.2478/aupcsm-2020-0010.

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AbstractThe n-tuples of commuting Hilbert space contractions are considered. We give a model of a commuting lifting of one contraction and investigate conditions under which a commuting lifting theorem holds for an n-tuple. A series of such liftings leads to an isometric dilation of the n-tuple. The method is tested on some class of triples motivated by Parrotts example. It provides also a new proof of the fact that a positive definite n-tuple has an isometric dilation.

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19

Marcantognini, S. A. M., and M. D. Morán. "A Schur type analysis of the minimal weak unitary Hilbert space dilations of a Kreĭn space bicontraction and the Relaxed Commutant Lifting Theorem in a Kreĭn space setting." Journal of Functional Analysis 259, no. 10 (November 2010): 2557–86. http://dx.doi.org/10.1016/j.jfa.2010.07.008.

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20

Gu, C. X. "On Causal Commutant Lifting Theorems." Journal of Mathematical Analysis and Applications 178, no. 2 (September 1993): 404–17. http://dx.doi.org/10.1006/jmaa.1993.1315.

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21

Popescu, Gelu. "A lifting theorem for symmetric commutants." Proceedings of the American Mathematical Society 129, no. 6 (October 31, 2000): 1705–11. http://dx.doi.org/10.1090/s0002-9939-00-05750-6.

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22

Frazho, A. E., S. ter Horst, and M. A. Kaashoek. "Coupling and Relaxed Commutant Lifting." Integral Equations and Operator Theory 54, no. 1 (October 1, 2005): 33–67. http://dx.doi.org/10.1007/s00020-005-1365-5.

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23

McCullough, Scott, and Saida Sultanic. "Ersatz Commutant Lifting with Test Functions." Complex Analysis and Operator Theory 1, no. 4 (June 25, 2007): 581–620. http://dx.doi.org/10.1007/s11785-007-0022-1.

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24

ter Horst, S. "Redheffer Representations and Relaxed Commutant Lifting." Complex Analysis and Operator Theory 5, no. 4 (February 9, 2010): 1051–72. http://dx.doi.org/10.1007/s11785-010-0046-9.

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25

McCullough, Scott. "Commutant lifting on a two holed domain." Integral Equations and Operator Theory 35, no. 1 (March 1999): 65–84. http://dx.doi.org/10.1007/bf01225528.

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26

Russo, Benjamin. "Lifting commuting 3-isometric tuples." Operators and Matrices, no. 2 (2017): 397–433. http://dx.doi.org/10.7153/oam-11-28.

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27

Russo, Benjamin. "Lifting commuting 3-isometric tuples." Operators and Matrices, no. 2 (2017): 397–433. http://dx.doi.org/10.7153/oam-2017-11-28.

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28

Ball, Joseph A. "Commutant lifting and interpolation: The time-varying case." Integral Equations and Operator Theory 25, no. 4 (December 1996): 377–405. http://dx.doi.org/10.1007/bf01203025.

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29

Biswas, Animikh. "A harmonic-type maximal principle in commutant lifting." Integral Equations and Operator Theory 28, no. 4 (December 1997): 373–81. http://dx.doi.org/10.1007/bf01309154.

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30

Stochel, Jan. "Lifting strong commutants of unbounded subnormal operators." Integral Equations and Operator Theory 43, no. 2 (June 2002): 189–214. http://dx.doi.org/10.1007/bf01200253.

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31

Ball, Joseph A., and Alexander Kheifets. "The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism." Integral Equations and Operator Theory 70, no. 1 (March 25, 2011): 17–62. http://dx.doi.org/10.1007/s00020-011-1873-4.

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32

Lee, Sang Hoon, Woo Young Lee, and Jasang Yoon. "Subnormality of Powers of Multivariable Weighted Shifts." Journal of Function Spaces 2020 (November 27, 2020): 1–11. http://dx.doi.org/10.1155/2020/5678795.

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Given a pair T ≡ T 1 , T 2 of commuting subnormal Hilbert space operators, the Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for the existence of a commuting pair N ≡ N 1 , N 2 of normal extensions of T 1 and T 2 ; in other words, T is a subnormal pair. The LPCS is a longstanding open problem in the operator theory. In this paper, we consider the LPCS of a class of powers of 2 -variable weighted shifts. Our main theorem states that if a “corner” of a 2-variable weighted shift T = W α , β ≔ T 1 , T 2 is subnormal, then T is subnormal if and only if a power T m , n ≔ T 1 m , T 2 n is subnormal for some m , n ≥ 1 . As a corollary, we have that if T is a 2-variable weighted shift having a tensor core or a diagonal core, then T is subnormal if and only if a power of T is subnormal.

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33

Müller, Vladimir. "Liftings and dilations of commuting systems of linear mappings on vector spaces." Operators and Matrices, no. 2 (2023): 567–72. http://dx.doi.org/10.7153/oam-2023-17-37.

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34

Bhattacharyya, Tirthankar, B. Krishna Das, and Haripada Sau. "Toeplitz Operators on the Symmetrized Bidisc." International Mathematics Research Notices, January 11, 2020. http://dx.doi.org/10.1093/imrn/rnz333.

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Abstract The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

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35

Didas, Michael. "A-Isometries and Hilbert-A-Modules Over Product Domains." Complex Analysis and Operator Theory 16, no. 5 (June 18, 2022). http://dx.doi.org/10.1007/s11785-022-01243-6.

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AbstractFor a compact set $$K \subset {\mathbb {C}}^n$$ K ⊂ C n , let $$A \subset C(K)$$ A ⊂ C ( K ) be a function algebra containing the polynomials $${\mathbb {C}}[z_1,\cdots ,z_n ]$$ C [ z 1 , ⋯ , z n ] . Assuming that a certain regularity condition holds for A, we prove a commutant-lifting theorem for A-isometries that contains the known results for isometric subnormal tuples in its different variants as special cases, e.g., Mlak (Studia Math. 43(3): 219–233, 1972) and Athavale (J. Oper. Theory 23(2): 339–350, 1990; Rocky Mt. J. Math. 48(1): 2018; Complex Anal. Oper. Theory 2(3): 417–428, 2008; New York J. Math. 25: 934–948, 2019). In the context of Hilbert-A-modules, our result implies the existence of an extension map "Equation missing" for hypo-Shilov-modules "Equation missing"$$(i=1,2)$$ ( i = 1 , 2 ) . By standard arguments, we obtain an identification "Equation missing" where "Equation missing" is the minimal $$C(\partial _A)$$ C ( ∂ A ) -extension of "Equation missing"$$(i=1,2)$$ ( i = 1 , 2 ) , provided that "Equation missing" is projective and "Equation missing" is pure. Using embedding techniques, we show that these results apply in particular to the domain algebra $$A=A(D)=C({\overline{D}})\cap {\mathcal {O}}(D)$$ A = A ( D ) = C ( D ¯ ) ∩ O ( D ) over a product domain $$D = D_1 \times \cdots \times D_k \subset {\mathbb {C}}^n$$ D = D 1 × ⋯ × D k ⊂ C n where each factor $$D_i$$ D i is either a smoothly bounded, strictly pseudoconvex domain or a bounded symmetric and circled domain in some $${\mathbb {C}}^{d_i}$$ C d i ($$1\le i \le k$$ 1 ≤ i ≤ k ). This extends known results from the ball and polydisc-case, Guo (Studia Math. 135(1): 1–12, 1999) and Chen and Guo (J. Oper. Theory 43: 69–81, 2000).

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36

Bisai, Bappa, Sourav Pal, and Prajakta Sahasrabuddhe. "On q-commuting co-extensions and q-commutant lifting." Linear Algebra and its Applications, November 2022. http://dx.doi.org/10.1016/j.laa.2022.11.003.

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37

Deepak, K. D., Deepak Kumar Pradhan, Jaydeb Sarkar, and Dan Timotin. "Commutant Lifting and Nevanlinna–Pick Interpolation in Several Variables." Integral Equations and Operator Theory 92, no. 3 (June 2020). http://dx.doi.org/10.1007/s00020-020-02582-9.

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38

Dolžan, David. "Invertible matrices over a class of semirings." Journal of Algebra and Its Applications, December 30, 2021. http://dx.doi.org/10.1142/s0219498823500792.

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We characterize the invertible matrices over a class of semirings such that the set of additively invertible elements is equal to the set of nilpotent elements. We achieve this by studying the liftings of the orthogonal sums of elements that are “almost idempotent” to those that are idempotent. Finally, we show an application of the obtained results to calculate the diameter of the commuting graph of the group of invertible matrices over the semirings in question.

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39

Frazho, A. E., M. A. Kaashoek, and F. van Schagen. "Solving Continuous Time Leech Problems for Rational Operator Functions." Integral Equations and Operator Theory 94, no. 3 (August 18, 2022). http://dx.doi.org/10.1007/s00020-022-02710-7.

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AbstractThe main continuous time Leech problems considered in this paper are based on stable rational finite dimensional operator-valued functions G and K. Here stable means that G and K do not have poles in the closed right half plane including infinity, and the Leech problem is to find a stable rational operator solution X such that $$\begin{aligned} G(s)X(s) = K(s) \quad (s\in \mathbb {C}_+) \quad \hbox {and}\quad \sup \{ \Vert X(s) \Vert :\Re s \ge 0 \} < 1 . \end{aligned}$$ G ( s ) X ( s ) = K ( s ) ( s ∈ C + ) and sup { ‖ X ( s ) ‖ : ℜ s ≥ 0 } < 1 . In the paper the solution of the Leech problem is given in the form of a state space realization. In this realization the finite dimensional operators involved are expressed in the operators of state space realizations of the functions G and K. The formulas are inspired by and based on ideas originating from commutant lifting techniques. However, the proof mainly uses the state space representations of the rational finite dimensional operator-valued functions involved. The solutions to the discrete time Leech problem on the unit circle are easier to develop and have been solved earlier; see, for example, Frazho et al. (Indagationes Math 25:250–274 2014).

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