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Journal articles on the topic 'Symmetrized bidisk'

Author: Grafiati

Published: 6 September 2023

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1

Bhattacharyya, Tirthankar, and Haripada Sau. "Interpolating sequences and the Toeplitz--Corona theorem on the symmetrized bidisk." Journal of Operator Theory 87, no. 1 (March 15, 2022): 435–59. http://dx.doi.org/10.7900/jot.2020oct07.2311.

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2

Bhattacharyya, Tirthankar, and Haripada Sau. "Holomorphic functions on the symmetrized bidisk – Realization, interpolation and extension." Journal of Functional Analysis 274, no. 2 (January 2018): 504–24. http://dx.doi.org/10.1016/j.jfa.2017.09.013.

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3

Agler, J., and N. J. Young. "Operators having the symmetrized bidisc as a spectral set." Proceedings of the Edinburgh Mathematical Society 43, no. 1 (February 2000): 195–210. http://dx.doi.org/10.1017/s0013091500020812.

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AbstractWe characterize those commuting pairs of operators on Hubert space that have the symmetrized bidisc as a spectral set in terms of the positivity of a hermitian operator pencil (without any assumption about the joint spectrum of the pair). Further equivalent conditions are that the pair has a normal dilation to the distinguished boundary of the symmetrized bidisc, and that the pair has the symmetrized bidisc as a complete spectral set. A consequence is that every contractive representation of the operator algebra A(Γ) of continuous functions on the symmetrized bidisc analytic in the interior is completely contractive. The proofs depend on a polynomial identity that is derived with the aid of a realization formula for doubly symmetric hereditary polynomials, which are positive on commuting pairs of contractions.

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4

Sarkar, Jaydeb. "Operator Theory on Symmetrized Bidisc." Indiana University Mathematics Journal 64, no. 3 (2015): 847–73. http://dx.doi.org/10.1512/iumj.2015.64.5541.

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5

Trybuła, Maria. "Invariant metrics on the symmetrized bidisc." Complex Variables and Elliptic Equations 60, no. 4 (August 28, 2014): 559–65. http://dx.doi.org/10.1080/17476933.2014.948543.

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6

COSTARA, C. "THE SYMMETRIZED BIDISC AND LEMPERT'S THEOREM." Bulletin of the London Mathematical Society 36, no. 05 (August 24, 2004): 656–62. http://dx.doi.org/10.1112/s0024609304003200.

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7

Pflug, Peter, and Włodzimierz Zwonek. "Exhausting domains of the symmetrized bidisc." Arkiv för Matematik 50, no. 2 (October 2012): 397–402. http://dx.doi.org/10.1007/s11512-011-0153-5.

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8

Bhattacharyya, Tirthankar, Anindya Biswas, and Anwoy Maitra. "On the geometry of the symmetrized bidisc." Indiana University Mathematics Journal 71, no. 2 (2022): 685–713. http://dx.doi.org/10.1512/iumj.2022.71.8896.

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9

Agler, Jim, Zinaida A. Lykova, and N. J. Young. "Extremal holomorphic maps and the symmetrized bidisc." Proceedings of the London Mathematical Society 106, no. 4 (October 26, 2012): 781–818. http://dx.doi.org/10.1112/plms/pds049.

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10

Agler, J., and N. J. Young. "A Schwarz Lemma for the Symmetrized Bidisc." Bulletin of the London Mathematical Society 33, no. 2 (March 2001): 175–86. http://dx.doi.org/10.1112/blms/33.2.175.

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11

Agler, J., and N. J. Young. "The hyperbolic geometry of the symmetrized bidisc." Journal of Geometric Analysis 14, no. 3 (September 2004): 375–403. http://dx.doi.org/10.1007/bf02922097.

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12

AGLER, J., and N. J. YOUNG. "THE COMPLEX GEODESICS OF THE SYMMETRIZED BIDISC." International Journal of Mathematics 17, no. 04 (April 2006): 375–91. http://dx.doi.org/10.1142/s0129167x06003564.

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We give formulae for all complex geodesics of the symmetrized bidisc G. There are two classes of geodesics: flat ones, indexed by the unit disc, and geodesics of degree 2, naturally indexed by G itself. The flat geodesics foliate G, and there is a unique geodesic through every pair of points of G. We also obtain a trichotomy result for left inverses of complex geodesics.

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13

Agler, Jim, and N. J. Young. "Realization of functions on the symmetrized bidisc." Journal of Mathematical Analysis and Applications 453, no. 1 (September 2017): 227–40. http://dx.doi.org/10.1016/j.jmaa.2017.04.003.

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14

Agler, Jim, Zinaida Lykova, and N. J. Young. "A geometric characterization of the symmetrized bidisc." Journal of Mathematical Analysis and Applications 473, no. 2 (May 2019): 1377–413. http://dx.doi.org/10.1016/j.jmaa.2019.01.027.

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15

Edigarian, Armen. "Proper holomorphic self-mappings of the symmetrized bidisc." Annales Polonici Mathematici 84, no. 2 (2004): 181–84. http://dx.doi.org/10.4064/ap84-2-8.

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16

Frosini, Chiara, and Fabio Vlacci. "A Julia's Lemma for the symmetrized bidisc 𝔾2." Complex Variables and Elliptic Equations 57, no. 10 (October 2012): 1121–34. http://dx.doi.org/10.1080/17476933.2010.534789.

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17

Agler, Jim, Zinaida A. Lykova, and N. J. Young. "3-Extremal Holomorphic Maps and the Symmetrized Bidisc." Journal of Geometric Analysis 25, no. 3 (July 15, 2014): 2060–102. http://dx.doi.org/10.1007/s12220-014-9504-3.

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18

Nikolov, Nikolai, Peter Pflug, and W{\l}odzimierz Zwonek. "An example of a bounded $\mathsf C$-convex domain which is not biholomorphic to a convex domain." MATHEMATICA SCANDINAVICA 102, no. 1 (March 1, 2008): 149. http://dx.doi.org/10.7146/math.scand.a-15056.

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We show that the symmetrized bidisc is a $\mathsf C$-convex domain. This provides an example of a bounded $\mathsf C$-convex domain which cannot be exhausted by domains biholomorphic to convex domains.

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19

PFLUG, PETER, and WLODZIMIERZ ZWONEK. "DESCRIPTION OF ALL COMPLEX GEODESICS IN THE SYMMETRIZED BIDISC." Bulletin of the London Mathematical Society 37, no. 04 (August 2005): 575–84. http://dx.doi.org/10.1112/s0024609305004418.

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20

Pal, Sourav, and Orr Moshe Shalit. "Spectral sets and distinguished varieties in the symmetrized bidisc." Journal of Functional Analysis 266, no. 9 (May 2014): 5779–800. http://dx.doi.org/10.1016/j.jfa.2013.12.022.

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21

Pal, Sourav, and Samriddho Roy. "A generalized Schwarz lemma for two domains related to μ-synthesis." Complex Manifolds 5, no. 1 (February 2, 2018): 1–8. http://dx.doi.org/10.1515/coma-2018-0001.

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AbstractWe present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock E. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc G2. In either case, our results generalize all previous results in this direction for E and G2.

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22

Agler, Jim, Zinaida Lykova, and N. J. Young. "Intrinsic Directions, Orthogonality, and Distinguished Geodesics in the Symmetrized Bidisc." Journal of Geometric Analysis 31, no. 8 (January 19, 2021): 8202–37. http://dx.doi.org/10.1007/s12220-020-00582-0.

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AbstractThe symmetrized bidisc $$\begin{aligned} G {\mathop {=}\limits ^\mathrm{{def}}}\{(z+w,zw):|z|<1,\quad |w|<1\}, \end{aligned}$$ G = def { ( z + w , z w ) : | z | < 1 , | w | < 1 } , under the Carathéodory metric, is a complex Finsler space of cohomogeneity 1 in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, G does not admit a natural notion of angle, but we nevertheless show that there is a notion of orthogonality. The complex tangent bundle TG splits naturally into the direct sum of two line bundles, which we call the sharp and flat bundles, and which are geometrically defined and therefore covariant under automorphisms of G. Through every point of G, there is a unique complex geodesic of G in the flat direction, having the form $$\begin{aligned} F^\beta {\mathop {=}\limits ^\mathrm{{def}}}\{(\beta +{\bar{\beta }} z,z)\ : z\in \mathbb {D}\} \end{aligned}$$ F β = def { ( β + β ¯ z , z ) : z ∈ D } for some $$\beta \in \mathbb {D}$$ β ∈ D , and called a flat geodesic. We say that a complex geodesic Dis orthogonal to a flat geodesic F if D meets F at a point $$\lambda $$ λ and the complex tangent space $$T_\lambda D$$ T λ D at $$\lambda $$ λ is in the sharp direction at $$\lambda $$ λ . We prove that a geodesic D has the closest point property with respect to a flat geodesic F if and only if D is orthogonal to F in the above sense. Moreover, G is foliated by the geodesics in G that are orthogonal to a fixed flat geodesic F.

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23

Tu, Zhenhan, and Shuo Zhang. "The Schwarz lemma at the boundary of the symmetrized bidisc." Journal of Mathematical Analysis and Applications 459, no. 1 (March 2018): 182–202. http://dx.doi.org/10.1016/j.jmaa.2017.10.061.

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24

Agler, Jim, Zinaida Lykova, and Nicholas Young. "Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc." Memoirs of the American Mathematical Society 258, no. 1242 (March 2019): 0. http://dx.doi.org/10.1090/memo/1242.

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25

Kosiński, Łukasz, and Włodzimierz Zwonek. "Nevanlinna–Pick Problem and Uniqueness of Left Inverses in Convex Domains, Symmetrized Bidisc and Tetrablock." Journal of Geometric Analysis 26, no. 3 (April 10, 2015): 1863–90. http://dx.doi.org/10.1007/s12220-015-9611-9.

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26

Chen, Liwei, Muzhi Jin, and Yuan Yuan. "Bergman Projection on the Symmetrized Bidisk." Journal of Geometric Analysis 33, no. 7 (April 20, 2023). http://dx.doi.org/10.1007/s12220-023-01263-4.

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27

DAS, B. KRISHNA, P. KUMAR, and H. SAU. "DETERMINING SETS FOR HOLOMORPHIC FUNCTIONS ON THE SYMMETRIZED BIDISK." Canadian Mathematical Bulletin, January 31, 2023, 1–13. http://dx.doi.org/10.4153/s0008439523000103.

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28

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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29

Jarnicki, Marek, and Peter Pflug. "On automorphisms of the symmetrized bidisc." Archiv der Mathematik 83, no. 3 (September 2004). http://dx.doi.org/10.1007/s00013-004-1049-4.

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30

Bhattacharyya, Tirthankar, B. Krishna Das, and Haripada Sau. "Addendum to: Toeplitz Operators on the Symmetrized Bidisc." International Mathematics Research Notices, September 13, 2021. http://dx.doi.org/10.1093/imrn/rnab252.

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31

Cho, Gunhee, and Yuan Yuan. "Bergman metric on the symmetrized bidisc and its consequences." International Journal of Mathematics, August 24, 2022. http://dx.doi.org/10.1142/s0129167x22500719.

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32

Agler, J., Z. Lykova, and N. Young. "Nonuniqueness of Carathéodory Extremal Functions on the Symmetrized Bidisc." Analysis Mathematica, April 20, 2022. http://dx.doi.org/10.1007/s10476-022-0138-6.

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33

Hamada, Hidetaka, and Hiroumi Segawa. "AN ELEMENTARY PROOF OF A SCHWARZ LEMMA FOR THE SYMMETRIZED BIDISC." Demonstratio Mathematica 36, no. 2 (April 1, 2003). http://dx.doi.org/10.1515/dema-2003-0208.

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34

Alshehri, Nujood M., and Zinaida A. Lykova. "A Schwarz Lemma for the Pentablock." Journal of Geometric Analysis 33, no. 2 (December 19, 2022). http://dx.doi.org/10.1007/s12220-022-01107-7.

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AbstractIn this paper, we prove a Schwarz lemma for the pentablock. The pentablock $$\mathcal {P}$$ P is defined by $$\begin{aligned} \mathcal {P}=\{(a_{21}, {\text {tr}}A, \det A) : A=[a_{ij}]_{i,j=1}^2 \in \mathbb {B}^{2\times 2}\} \end{aligned}$$ P = { ( a 21 , tr A , det A ) : A = [ a ij ] i , j = 1 2 ∈ B 2 × 2 } where $$\mathbb {B}^{2\times 2}$$ B 2 × 2 denotes the open unit ball in the space of $$2\times 2$$ 2 × 2 complex matrices. The pentablock is a bounded non-convex domain in $$\mathbb {C}^3$$ C 3 which arises naturally in connection with a certain problem of $$\mu $$ μ -synthesis. We develop a concrete structure theory for the rational maps from the unit disc $$\mathbb {D}$$ D to the closed pentablock $$\overline{\mathcal {P}}$$ P ¯ that map the unit circle $$\mathbb {T}$$ T to the distinguished boundary $$b\overline{\mathcal {P}}$$ b P ¯ of $$\overline{\mathcal {P}}$$ P ¯ . Such maps are called rational $${\overline{\mathcal {P}}}$$ P ¯ -inner functions. We give relations between $${\overline{\mathcal {P}}}$$ P ¯ -inner functions and inner functions from $$\mathbb {D}$$ D to the symmetrized bidisc. We describe the construction of rational $${\overline{\mathcal {P}}}$$ P ¯ -inner functions $$x = (a, s, p) : \mathbb {D} \rightarrow \overline{\mathcal {P}}$$ x = ( a , s , p ) : D → P ¯ of prescribed degree from the zeroes of a, s and $$s^2-4p$$ s 2 - 4 p . The proof of this theorem is constructive: it gives an algorithm for the construction of a family of such functions x subject to the computation of Fejér–Riesz factorizations of certain non-negative trigonometric functions on the circle. We use properties and the construction of rational $${\overline{\mathcal {P}}}$$ P ¯ -inner functions to prove a Schwarz lemma for the pentablock.

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