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Journal articles on the topic 'Bidisc'

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Published: 6 September 2023

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1

Bandura, A. I., and N. V. Petrechko. "Properties of power series of analytic in a bidisc functions of bounded $\mathbf{L}$-index in joint variables." Carpathian Mathematical Publications 9, no. 1 (June 8, 2017): 6–12. http://dx.doi.org/10.15330/cmp.9.1.6-12.

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We generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic in a bidisc functions, where $\mathbf{L}(z)=(l_1(z_1,z_2),$ $l_{2}(z_1,z_2)),$ $l_j:\mathbb{D}^2\to \mathbb{R}_+$ is a continuous function, $j\in\{1,2\},$ $\mathbb{D}^2$ is a bidisc $\{(z_1,z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1\}.$ The propositions describe a behaviour of power series expansion on a skeleton of a bidisc. We estimated power series expansion by a dominating homogeneous polynomial with the degree that does not exceed some number depending only from radii of bidisc. Replacing universal quantifier by existential quantifier for radii of bidisc, we also proved sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables for analytic functions which are weaker than necessary conditions.

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2

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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3

Baksa, Vitalina, Andriy Bandura, and Oleg Skaskiv. "Analogs of Fricke's theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 33 (December 27, 2019): 16–26. http://dx.doi.org/10.37069/1683-4720-2019-33-1.

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In this paper, we present necessary and sufficient conditions of boundedness of $\mathbb{L}$-index in joint variables for vector-functions analytic in the unit ball, where $\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$ is a positive continuous vector-function, $\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$ Particularly, we deduce analog of Fricke's theorems for this function class, give estimate of maximum modulus on the skeleton of bidisc. The first theorem concerns sufficient conditions. In this theorem we assume existence of some radii, for which the maximum of norm of vector-function on the skeleton of bidisc with larger radius does not exceed maximum of norm of vector-function on the skeleton of bidisc with lesser radius multiplied by some costant depending only on these radii. In the second theorem we show that boundedness of $\mathbf{L}$-index in joint variables implies validity of the mentioned estimate for all radii.

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4

Nakazi, Takahiko. "Szegö's Theorem and Uniform Algebras." Canadian Mathematical Bulletin 54, no. 2 (June 1, 2011): 338–46. http://dx.doi.org/10.4153/cmb-2011-017-4.

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5

Nakazi, Takahiko. "Szego's Theorem on a Bidisc." Transactions of the American Mathematical Society 328, no. 1 (November 1991): 421. http://dx.doi.org/10.2307/2001888.

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6

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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7

Nakazi, Takahiko. "Szegő’s theorem on a bidisc." Transactions of the American Mathematical Society 328, no. 1 (January 1, 1991): 421–32. http://dx.doi.org/10.1090/s0002-9947-1991-1028762-2.

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8

Kosiński, Łukasz, Pascal Thomas, and Włodzimierz Zwonek. "Coman conjecture for the bidisc." Pacific Journal of Mathematics 287, no. 2 (March 9, 2017): 411–22. http://dx.doi.org/10.2140/pjm.2017.287.411.

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9

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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10

Lu, Yufeng. "Commuting of Toeplitz operators on the Bergman spaces of the bidisc." Bulletin of the Australian Mathematical Society 66, no. 2 (October 2002): 345–51. http://dx.doi.org/10.1017/s0004972700040181.

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11

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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12

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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13

Hedenmalm, Håkan. "Closed ideals in the bidisc algebra." Arkiv för Matematik 28, no. 1-2 (December 1990): 111–17. http://dx.doi.org/10.1007/bf02387368.

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14

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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15

Nakazi, Takahiko. "Invariant subspaces in the bidisc and commutators." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 56, no. 2 (April 1994): 232–42. http://dx.doi.org/10.1017/s1446788700034856.

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AbstractLet M be an invariant subspace of L2 (T2) on the bidisc. V1 and V2 denote the multiplication operators on M by coordinate functions z and ω, respectively. In this paper we study the relation between M and the commutator of V1 and , For example, M is studied when the commutator is self-adjoint or of finite rank.

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16

Cheung, Wing Sum, Siqi Fu, Steven G. Krantz, and Bun Wong. "A smoothly bounded domain in a complex surface with a compact quotient." MATHEMATICA SCANDINAVICA 91, no. 1 (September 1, 2002): 82. http://dx.doi.org/10.7146/math.scand.a-14380.

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We study the classification of smoothly bounded domains in complex manifolds that cover compact sets. We prove that a smoothly bounded domain in a hyperbolic complex surface that covers a compact set is either biholomorphic to the ball or covered by the bidisc.

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17

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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18

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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19

Lin, Kai-Ching. "Interpolation between Hardy spaces on the bidisc." Studia Mathematica 84, no. 1 (1986): 89–96. http://dx.doi.org/10.4064/sm-84-1-89-96.

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20

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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21

IZUCHI, Keiji, and Takahiko NAKAZI. "Backward shift invariant subspaces in the bidisc." Hokkaido Mathematical Journal 33, no. 1 (February 2004): 247–54. http://dx.doi.org/10.14492/hokmj/1285766003.

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22

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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23

Pipher, Jill, and Lesley A. Ward. "BMO from dyadic BMO on the bidisc." Journal of the London Mathematical Society 77, no. 2 (February 26, 2008): 524–44. http://dx.doi.org/10.1112/jlms/jdm114.

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24

Nakazi, Takahiko. "Norms of Hankel operators on a bidisc." Proceedings of the American Mathematical Society 108, no. 3 (March 1, 1990): 715. http://dx.doi.org/10.1090/s0002-9939-1990-1000162-5.

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25

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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26

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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27

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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28

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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29

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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30

Wong, B. "Characterization of the Bidisc by Its Automorphism Group." American Journal of Mathematics 117, no. 2 (April 1995): 279. http://dx.doi.org/10.2307/2374914.

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31

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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32

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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33

Nakazi, Takahiko. "Extremal problems inH1for continuous kernels on a bidisc." Complex Variables and Elliptic Equations 58, no. 9 (September 2013): 1321–29. http://dx.doi.org/10.1080/17476933.2012.667407.

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34

Agler, Jim, John E. McCarthy, and N. J. Young. "Facial behaviour of analytic functions on the bidisc." Bulletin of the London Mathematical Society 43, no. 3 (February 26, 2011): 478–94. http://dx.doi.org/10.1112/blms/bdq115.

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35

Redett, David, and James Tung. "Invariant subspaces in Bergman space over the bidisc." Proceedings of the American Mathematical Society 138, no. 07 (July 1, 2010): 2425. http://dx.doi.org/10.1090/s0002-9939-10-10337-2.

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36

Cheng, Raymond. "Weakly and strongly outer functions on the bidisc." Michigan Mathematical Journal 39, no. 1 (1992): 99–109. http://dx.doi.org/10.1307/mmj/1029004458.

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37

Hedenmalm, Håkan. "Outer functions in function algebras on the bidisc." Transactions of the American Mathematical Society 306, no. 2 (February 1, 1988): 697. http://dx.doi.org/10.1090/s0002-9947-1988-0933313-4.

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38

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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39

Nakazi, Takahiko. "Slice Maps and Multipliers of Invariant Subspaces." Canadian Mathematical Bulletin 39, no. 2 (June 1, 1996): 219–26. http://dx.doi.org/10.4153/cmb-1996-028-4.

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AbstractLet be the closed bidisc and T2 be its distinguished boundary. For be a slice map, that is, and Then ker Φαβ is an invariant subspace, and it is not difficult to describe ker Φαβ and In this paper, we study the set of all multipliers for an invariant subspace M such that the common zero set of M contains that of ker Φαβ.

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40

Nakazi, Takahiko. "Errata to "Norms of Hankel Operators on a Bidisc"." Proceedings of the American Mathematical Society 115, no. 3 (July 1992): 873. http://dx.doi.org/10.2307/2159240.

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41

Nakazi, Takahiko. "Errata to: ‘‘Norms of Hankel operators on a bidisc”." Proceedings of the American Mathematical Society 115, no. 3 (March 1, 1992): 873. http://dx.doi.org/10.1090/s0002-9939-1992-1072346-3.

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42

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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43

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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44

Nakazi, Takahiko. "Toeplitz Operators And Weighted Norm Inequalities On The Bidisc." Mathematical Inequalities & Applications, no. 3 (2001): 429–41. http://dx.doi.org/10.7153/mia-04-38.

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45

Nakazi, Takahiko. "Exposed points and extremal problems inH1 on a bidisc." Mathematische Nachrichten 258, no. 1 (September 2003): 97–104. http://dx.doi.org/10.1002/mana.200310089.

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46

Blower, Gordon. "Bilinear forms on vector Hardy spaces." Glasgow Mathematical Journal 39, no. 3 (September 1997): 371–78. http://dx.doi.org/10.1017/s0017089500032286.

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AbstractLet φ: ℋ → be a bilinear form on vector Hardy space. Introduce the symbol φ of Φ by (φ (Z1, Z2), a ⊗ b) = Φ (K21 ⊗ a, K22 ⊗ b ), where Kw is the reproducing kernel for w ∈ D. We show that Φ extends to a bounded bilinear form on provided that the gradient defines a Carleson measure in the bidisc D2. We obtain a sufficient condition for Φ to extend to a Hilbert space. For vectorial bilinear Hankel forms we obtain an analogue of Nehari's Theorem.

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47

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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48

Cegrell, Urban. "On ideals generated by bounded analytic functions in the bidisc." Bulletin de la Société mathématique de France 121, no. 1 (1993): 109–16. http://dx.doi.org/10.24033/bsmf.2202.

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49

Wang, Wei. "A discrete transform and Triebel-Lizorkin spaces on the bidisc." Transactions of the American Mathematical Society 347, no. 4 (April 1, 1995): 1351–64. http://dx.doi.org/10.1090/s0002-9947-1995-1254857-8.

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50

Nakazi, Takahiko. "Certain Invariant Subspaces of H2 and L2 on a Bidisc." Canadian Journal of Mathematics 40, no. 5 (October 1, 1988): 1272–80. http://dx.doi.org/10.4153/cjm-1988-055-6.

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We let T2 be the torus that is the cartesian product of 2 unit circles in C. The usual Lebesgue spaces, with respect to the Haar measure m of T2, are denoted by Lp = Lp(T2), and Hp = Hp(T2) is the space of all f in LP whose Fourier coefficientsare 0 as soon as at least one component of (j, ℓ) is negative.A closed subspace M of L2 is said to be invariant ifWhenever this is the case, it follows that fM ⊂ M for every f in H∞. One can ask for a classification or an explicit description (in some sense) of all invariant subspaces of L2, but this seems out of reach.

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