ABSTRACTS. Operator Theory and Its Applications

Transcription

1 Volume 17(2018) ABSTRACTS Operator Theory and Its Applications July 18 19, 2018 The K-Hotel Gyeongju, Gyeongju, Korea Supported by National Research Foundation of Korea Seoul National University - Department of Mathematical Sciences - BK21 PLUS SNU Mathematical Sciences Division - The Research Institute of Mathematics

2 TABLE OF CONTENTS Reducibility of the power of a C 0 (1) operator Caixing Gu 1 Hyponormal singular integral operators and Toeplitz operators Takanori Yamamoto 2 Difficulty on the existence of canonical system flows Injo Hur 3 Toeplitz operators with symbols of Borel measures Jashui Park 4 Semigroup C -algebras of non-amenable semigroups Sun Young Jang 5 Normality, self-adjointness and isometry of truncated singular integral operators Dong-O Kang 6 Hamburger-type weighted shifts Joo Young Jin 7 Conjugation on a Banach space Haruna Motoyoshi and Muneo Chō 8 On properties of (m, C)-isometric commuting tuples of operators Ji Eun Lee 9 10 A subnormal completion problem for weighted shifts on directed trees Il Bong Jung 11

3 Toeplitz operators on the Fock spaces Jongrak Lee 12 Semicircular elements induced by orthogonal projections Ilwoo Cho 13 Symmetric properties and operator transforms Chafiq Benhida 14 The powers of quasinormal operators of a commuting pair Jasang Yoon 15

4 Volume 17(2018), 1 1 Reducibility of the power of a C 0 (1) operator Caixing Gu California Polytechnic State University, U.S.A. cgu@calpoly.edu Inspired by the work of Douglas and Foias in 2006 on the structure of the square of a C 0 (1) operator, we form a conjecture about a certain reducibility of any power N of a C 0 (1) operator. We then prove this conjecture for N = 3 by determining explicitly the relevant reducing subspaces.

5 Volume 17(2018), 2 2 Hyponormal singular integral operators and Toeplitz operators Takanori Yamamoto Hokkai-Gakuen University, Japan yamamoto@elsa.hokkai-s-u.ac.jp Let α and β be functions in L (T), where T is the unit circle. Let P denote the orthogonal projection from L 2 (T) onto the Hardy space H 2 (T), and Q = I P, where I is the identity operator on L 2 (T). In this talk, we consider the hyponormality of singular integral operators S α,β on L 2 (T) of the form S α,β f = αp f + βqf, for f L 2 (T), and Toeplitz operators on H 2 (T) of the form T α f = P (αf), for f H 2 (T). This is a joint work with Professor Takahiko Nakazi. References [1] C.C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103(1988), [2] I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations, Vol. 1, Birkhäuser, Basel, [3] C. Gu, Algebraic properties of Cauchy singular integral operators on the unit circle, Taiwanese J. Math. 20 (2016), [4] C. Gu, I.S. Hwang, D. Kang and W.Y. Lee, Normal singular Cauchy integral operators with operator-valued symbols, J. Math. Anal. Appl. 447(2017), [5] I.S. Hwang and W.Y. Lee, Hyponormal Toeplitz operators with rational symbols, J. Oper. Theory 56 (2006), [6] T. Ito and T.K. Wong, Subnormality and quasinormality of Toeplitz operators, Proc. A.M.S. 34 (1972), [7] Y. Kim, E. Ko, J. Lee and T. Nakazi, Hyponormality of singular Cauchy integral operators with matrix-valued symbols, preprint. [8] R.A. Martínez-Avendaño and P. Rosenthal, An introduction to operators on the Hardy-Hilbert space, Springer, [9] T. Nakazi, Range inclusion of two same type concrete operators, preprint. [10] T. Nakazi, Hyponormal singular integral operators with Cauchy kernel on L 2, preprint. [11] T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), [12] T. Nakazi and T. Yamamoto, Normal singular integral operators with Cauchy kernel on L 2, Integral Equations Operator Theory 78 (2014), [13] N.K. Nikolski, Operators, functions, and systems: An Easy Reading. Vol. 1, Amer. Math. Soc., Providence, [14] D. Sarason, Generalized interpolation in H, Trans. Amer. Math. Soc. 127 (1967), [15] T. Yamamoto, Majorization of singular integral operators with Cauchy kernel on L 2, Ann. Funct. Anal. 5 (2014),

6 Volume 17(2018), 3 3 Difficulty on the existence of canonical system flows Injo Hur UNIST, Korea injohur@unist.ac.kr We will revisit the structure of Korteweg-de Vries (KdV) flows, especially KdV hierarchy, whose spectra are invariant, when the time passes. This iso-spectral flows are related to the existence of solitary waves like Tsunami. Except a traveling wave which is the simplest example, KdV equation is the most well-studied one having such solutions. Since KdV equations are naturally generalized to canonical systems, we may expect that iso-spectral canonical systems would be found. In this talk, we will, however, show that it will be very difficult for such canonical flows like KdV ones to exist. To see this, due to the lack of existence on operators related to canonical systems, so-called zero-curvature equation will be applied, which is equivalent to the famous Lax pair formalism.

7 Volume 17(2018), 4 4 Toeplitz operators with symbols of Borel measures Jaehui Park Seoul National University, Korea nephenjia@snu.ac.kr In this talk, I consider Toeplitz operators T µ with symbols of complex Borel measures µ, and give a characterization of bounded Toeplitz operators: T µ is bounded if and only if µ is a T-Carleson measure when the domain of T µ contains all polynomials. This is based on a joint work with Caixing Gu.

8 Volume 17(2018), 5 5 Semigroup C -algebras of non-amenable semigroups Sun Young Jang University of Ulsan, Korea jsym@ulsan.ac.kr The study of semigroup C -algebras has started with the question whether the theory of group C -algebras can be extended to more basic cases, namely semigroups. The starting result was made by L.A. Coburn, which states all C -algebras generated by single non-unitary isometry are isomorphic. The generalization of Coburn s result is called the uniqueness property of C -algebras generated by isometric representations of semigroups. A. Nica has introduced quasi-lattice ordered groups that could make the theory of the uniqueness property advance to a more interesting state. The uniqueness property of semigroup C -algebras is related with the amenability of semigroups as the amenability of locally compact groups is related with the isomorphism of group C -algebras. Xin Li has made a different construction of semigroup C -algebras to enlarge the amenability of semigroups. We show that how much the uniqueness property is related with the order structure of semigroups and also the structure of semigroup C -algebras of some non-quasi lattice ordered semigroups.

9 Volume 17(2018), 6 6 Normality, self-adjointness and isometry of truncated singular integral operators Dong-O Kang Chungnam National University, Korea dokang@cnu.ac.kr For truncated singular integral operators on the unit circle, self-adjointness, isometry, coisometry and normality will be characterized. These characterizations will be complete except for the case of self-adjointness.

10 Volume 17(2018), 7 7 Hamburger-type weighted shifts Joo Young Jin Kyungpook National University, Korea pss9611@hanmail.net We indicate how our subject emerges from the confluence of several streams of analysis, including the classical moment problems, the theory of positive matrices and subnormal operator theory. Some new properties H(n) (n = 1, 2,...) and a Hamburger-type weighted shift are considered via a Hamburger moment sequence. We discuss examples to show the various H(n) are distinct; study flatness, backward n-step extensions and perturbations of weighted shifts; and, given three initial weights α 0, α 1, α 2 with α 0 α 2 < α 1, we produce a completion: a weighted shift of Hamburger-type but not subnormal, extending a (subnormal) completion by Stampfli in the case α 0 < α 1 < α 2.

11 Volume 17(2018), 8 8 Conjugation on a Banach space Haruna Motoyoshi 1 and Muneo Chō 2 Kanagawa University, Japan 1 r cv@jindai.jp; 2 chiyom01@kanagawa-u.ac.jp Let X be a complex Banach space and C be an operator on X. C is said to be a conjugation if C 2 = I, C 1, C(x + y) = Cx + Cy, C(λ x) = λ x ( x, y X, λ C). From the definition, it holds C = 1. We have a talk about spectral properties of a bounded linear operator T B(X ) and others concerning with a conjugation C on a Banach space X. References 1. G. de Barra, Some algebras of operators with closed convex numerical range, Proc. Amer. Math. Soc. A 72(1972), M. Barraa and V. Muller, On the essential numerical range, Acta Sci. Math. (Szeged) 71(2005), F.F. Bonsal and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras, London Math. Soc. Lecture Note Series, 2, F.F. Bonsal and J. Duncan, Numerical ranges II, London Math. Soc. Lecture Note Series, 10, M. Chō and K. Tanahashi, On conjugations for Banach spaces, Sci. Math. Jpn. 6. R.S. Garcia and M. Putinar, Complex symmetric operators and applications, Trans. Amer. Math. Soc. 358(2005), R.S. Garcia and M. Putinar, Complex symmetric operators and applications II, Trans. Amer. Math. Soc. 359(2007), S. Jung, E. Ko and Ji Eun Lee, On complex symmetric operator matrices, J. Math. Anal. Appl., 406(2013), K. Mattila, Normal operators and proper boundary points of operators on Banach space, Ann. Acad. Sci. Fenn. Ser. A ID Math. Dessertationes 19, B.-Y. Shaw, On numerical ranges of generalized derivations and related properties, J. Austral. Math. Soc. 36(1984),

12 Volume 17(2018), 9 10 On properties of (m, C)-isometric commuting tuples of operators Ji Eun Lee Sejong University, Korea jieunlee7@sejong.ac.kr; jieun7@ewhain.net A commuting tuples of operators T = (T 1,, T d ) B (d) (H) is said to be (m, C)- isometric tuple if Q m (T) := ( )( m ) ( 1) m k k! k β! T β CT β C = 0 0 k m for some positive integer m and some conjugation C. In this paper, we study the class of (m, C)-isometries for tuple of commuting operators which is a generalization of the class of (m, C)-isometric operators. We concentrate on a multivariable generalization of these single variable (m, C)-isometric operators and give some of their basic properties of such operators. The talk is based on joint work with Ould Ahmed Mahmoud Sid Ahmed and Muneo Chō. β =k References [1] J. Agler and M. Stankus, m-isometric transformations of Hilbert space I, Integr. Equat. Oper. Theory, 21(1995), [2] C. Ambrozie, M. Englis and V. Müller, Operator tuples and analytic models over general domains in C n, J. Operator Theory, 47(2002), [3] A. Ben Amor, An extension of Henrici theorem for the joint approximate spectrum of commuting spectral operators, J. Aust. Math. Soc., 75(2003), [4] T. Bermúdez, A. Martinón and V. Müller, (m, q)-isometries on metric spaces, J. Operator Theory, 72(2) (2014), [5] M. Chō, R.E. Curto and T. Huruya, n-tuples of operators satisfying σ T (AB) = σ T (BA), Linear Algebra Appl., 341(2002), [6] M. Chō, I.H. Jeon, I.B. Jung, J.I. Lee, and K. Tanahashi, Joint spectra of n-tuples of generalized Aluthge transformations, Rev. Roumaine Math. Pures Appl., 46(6) (2001), [7] M. Chō, I.H. Jeon and J.I. Lee, Joint spectra of doubly commuting n-tuples of operators and their Aluthge transforms, Nihonkai Math. J., 11(1) (2000), [8] M. Chō, E. Ko and J. E. Lee, On (m, C)-isometric operators, Complex Anal. Oper. Theory, 10 (2016),

13 [9] M. Chō, E. Ko and J. E. Lee, (, C)-isometric operators, Operators and Matrices, 113 (2017), [10] M. Chō, J. E. Lee and H. Motoyoshi, On [m, C]-isometric operators, Filomat 31:7 (2017), [11] M. Chō and V. Müller, Spectral commutativity of multioperators, Functional Analysis, Approximation and Computation, 4:1 (2012), [12] M. Chō and W. Żelazko, On geometric spectral radius of commuting n-tuples of operators, Hokkaido Math. J., 21(2)(1992), [13] P. H. W. Hoffmann and M. Mackey, (m, p)-and (m, )-isometric operator tuples on normed spaces, Asian-Eur. J. Math., 8(2) (2015), [14] J. Gleason and S. Richter, m-isometric commuting tuples of operators on a Hilbert space, Integr. Equat. Oper. Theory, 56(2) (2006), [15] C. Gu, The (m; q)-isometric weighted shifts on l p spaces, Integral Equations Operator Theory, 82 (2015) [16] O. A. Mahmoud Sid Ahmed, M. Chō and J. E. Lee, On n-quasi-(m, C)-isometric operators, preprint. [17] Z. Slodkowski and W. Żelazko, On joint spectra of commuting families of operators, Studia Math., 50(1974), This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2016R1A2B ).

14 Volume 17(2018), A subnormal completion problem for weighted shifts on directed trees Il Bong Jung Kyungpook National University, Korea ibjung@knu.ac.kr Given a directed tree and a collection of weights on a subtree, the subnormal completion problem is to determine whether the weights may be completed to the weights of an injective, bounded, subnormal weighted shift on the Hilbert space arising from the full tree. We study this problem (which generalizes significantly the classical subnormal completion problem for weighted shifts) both from a measure-theoretic point of view and in terms of initial data, for various classes of trees with a single branching point. We give several characterizations of when such a completion is possible. Considered also are connections with Stieltjes moment sequences, flatness of a completion, completions in which the resulting measures may be taken to be finitely atomic, and we provide a result showing that in certain circumstances the present completion problem is equivalent to a related classical completion problem. (This is a joint work with G. Exner, J. Stochel, and H. Y. Yun.)

15 Volume 17(2018), Toeplitz operators on the Fock spaces Jongrak Lee Ewha Womans University, Korea In this talk, we consider the properties of Toeplitz operators on the Fock spaces F 2. Next, we characterize the necessary and sufficient conditions for the hyponormality of Toeplitz operators T ϕ on F 2 with polynomial symbol ϕ. Moreover, we consider the necessary condition for hyponormality of T ϕ with trigonometric polynomial symbol ϕ. References [1] Cowen, C: Hyponormality of Toeplitz operators. Proc. Amer. Math. Soc. 103, (1988) [2] Curto, RE, Hwang, IS, Lee, WY: Hyponormality and subnormality of block Toeplitz operators. Adv. Math. 230, (2012) [3] Curto, RE, Lee, WY: Joint hyponormality of Toeplitz pairs. Memoirs Amer. Math. Soc. 712, (2001) [4] Hwang, IS: Hyponormal Toeplitz operators on the Bergman spaces. J. Korean Math. Soc. 42, (2005) [5] Hwang, IS, Kim, IH, Lee, WY: Hyponormality of Toeplitz operators with polynomial symbol. Math. Ann. 313, (1999) [6] Hwang, IS, Lee, WY: Hyponormality of trigonometric Toeplitz operators. Trans. Amer. Math. 354, (2002) [7] Nakazi, T, Takahashi, K: Hyponormal Toeplitz operators and extremal problems of Hardy spaces. Trans. Amer. Soc. 338, (1993) [8] Sadraoui, H: Hyponormality of Toeplitz operators and Composition operators. Thesis, Purdue University 1992 [9] Stroethoff, S: Compact Toeplitz operators on generalized Fock spaces. Acta Sci. Math. (Szeged) 64, (1998) [10] Stroethoff, K: Hankel and Toeplitz operators on the Fock space. Mich. Math. J. 39(1), 3 6 (1992) [11] Zhu, K: Analysis on Fock spaces. Springer, New York (2012)

16 Volume 17(2018), Semicircular elements induced by orthogonal projections Ilwoo Cho St. Ambrose University, USA In this talk, we consider how to construct semicircular elements induced by mutually orthogonal integer-many projections in a certain C -probability space. As application, we consider the weighted-semicircular laws and the semicircular law from p-adic analysis on p-adic number fields over primes p.

17 Volume 17(2018), Symmetric properties and operator transforms Chafiq Benhida Université de Lille I, France The class of symmetric operators is a quite large class. It contains many standard operators such as normal operators, Hankel matrices, finite Toeplitz matrices, all truncated Toeplitz operators, and Volterra integration operators. We ll present here some recent results on some transforms of symmetric operators.

18 Volume 17(2018), The powers of quasinormal operators of a commuting pair Jasang Yoon University of Texas Rio Grande Valley, USA jasang.yoon@utrgv.edu In this talk we investigate some properties of jointly and spherically quasinormal operators for a commuting pair. This is a joint work with Professors Raul Curto and Sang Hoon Lee