Abstract 22, June (Saturday) • 10:00–10:30 Ilya M. Spitkovsky ([email protected] and [email protected]) Mathematics Program

Abstract

22, June (Saturday)

• 10:00–10:30 Ilya M. Spitkovsky ([email protected] and [email protected]) Mathematics Program, New York University Abu Dhabi (NYUAD), UAE Title: Inverse continutiy of the numerical range map

Abstract: Let A be a linear bounded operator acting on a Hilbert space H. The numerical range W (A) of A can be thought of as the image of the units sphere of H under the numerical range generating function fA : x 7→ (Ax, x). This talk is devoted to continuity −1 properties of the (multivalued) inverse mapping fA . In particular, strong continuity of −1 fA on the interior of W (A) is established. Co-author: Brian Lins (Hampden-Sydney College, Virginia, USA).

• 10:30–11:00 Paul Reine Kennett L. Dela Rosa ([email protected]) Department of Mathematics, Drexel University, USA Title: Location of Ritz values in the numerical range of normal matrices

Abstract: In 2013, Carden and Hansen proved that ﬁxing µ1 ∈ W (A)\∂W (A), where A ∈ 3×3 C is a normal matrix with noncollinear eigenvalues, determines a unique number µ2 ∈ W (A) so that {µ1, µ2} forms a 2-Ritz set for A. They recognized that µ2 is the isogonal conjugate of µ1 with respect to the triangle formed by connecting the three eigenvalues of A. In this talk, we consider the analogous problem for a 4-by-4 normal matrix A. In particular, given µ1 ∈ W (A) in the interior of one of the quadrants formed by the diagonals of W (A), we prove that if {µ1, µ2} forms a 2-Ritz set, then µ2 lies in the convex hull of two eigenvalues of A and the two isogonal conjugates of µ1 with respect to the two triangles containing µ1. We examine how such a result can be used to understand 2-Ritz sets of n-by-n normal matrices. Co-author(s): Hugo J. Woerdeman (Department of Mathematics, Drexel University, USA).

1 • 11:15–11:45 Pan-Shun Lau ([email protected]) University of Nevada, Reno, USA Title: The decomposable numerical range of derivations Abstract: Let 1 ≤ k ≤ n be positive integers, G be a subgroup of the symmetric group of order k and χ be an irreducible character of G. The kth generalized numerical range of A ∈ Cn×n associated with G and χ is deﬁned by { } G G ∗ ∈ Cn×k ∗ Wχ (A) = dχ (V AV ): V ,V V = Ik ,

G where dχ is the generalized matrix function associated with G and χ. It is closely related to the decomposable numerical range of derivations. When k = 1, it reduces to the classical G numerical range of A. In the talk, we shall discuss the geometric properties of Wχ (A) such as the convexity and star-shapedness. Co-authors: Nung-Sing Sze (PolyU); Chi-Kwong Li (W&M).

2 • 11:45–12:15 Muneo Ch¯o ([email protected]) Department of Mathematics, Kanagawa University, Japan Title: Numerical ranges of Banach space operators Abstract: Let X be a complex Banach space and X ∗ be the dual space of X . For a bounded linear operator T on X , let the numerical range V (T ) of T is given by

V (T ) = {f(T x):(x, f) ∈ Π },

where Π is deﬁned by Π = { (x, f) ∈ X × X ∗ : ∥f∥ = f(x) = ∥x∥ = 1 }.

In this talk I’ll introduce properties of numerical ranges of T .

References

[1] M. Barra and V. M¨uller,On the essential numerical range, Acta Sci. Math. (Szeged) 71 (2005), 285-298. [2] 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, 1971. [3] F.F. Bonsal and J. Duncan, Numerical ranges II, London Math. Soc. Lecture Note Series. 10, 1973. [4] M. Ch¯o,Semi-normal operators on uniformly smooth Banach spaces, Glasgow Math. J. 32 (1990), 273-276. [5] M. Ch¯o,Hyponormal operators on uniformly smooth spaces, J. Austral. Math. Soc. 50 (1991), 594-598. [6] M. Ch¯o,Hyponormal operators on uniformly convex spaces, Acta Sci. Math. (Szeged) 55 (1991), 141-147. [7] M. Ch¯oand T. Huruya, A remark on numerical range of semi-hyponormal operators, LMLA, 58 (2010), 711-714. [8] M. Ch¯oand K. Tanahashi, On conjugations for Banach spaces, Sci. Math. Jpn. 81 (2018), 37-45. [9] T. Furuta, Introduction to linear operators, Taylor and Francis, 2001. [10] S. Jung, E. Ko and J. E. Lee, On complex symmetric operators, J. Math. Anal. Appl. 406 (2013), 373-385. [11] H. Motoyoshi, Linear operators and conjugations on a Banach space, to appear in Acta Sci. Math. (Szged).

3 • 14:00–14:30 Hiroshi Nakazato ([email protected]) Department of Mathematics and Physics, Hirosaki Universit, Japan Title: Period matrix of a Riemann surface and the numerical range Abstract: We present a method to compute the period matrix of a Riemann surfacce arising as the boundary generating curve of the numerical range of a matrix. Co-author(s): This talk is based on some joint worls with Professor Mao-Ting Chien.

• 14:30–15:00 Mao-Ting Chien ([email protected]) Department of Mathematics, Soochow University, Taiwan Title: Computing the diameter and width of the numerical range Abstract: The diameter(resp. width) of the numerical range of a matrix is deﬁned to be the largest(resp. smallest) distance of two parallel lines tangent to its boundary. The boundary curve of a numerical range is called a curve of constant width if its diameter and width are equal. In this talk, we provide an algorithm for computing the diameter and width of the numerical range, formulate the diameter of the numerical range of some unitary bordering matrices, and determine the condition for the boundary of the numerical range of certain Toeplitz matrices to be a curve of constant width. Co-author(s): Hiroshi Nakazato, Jie Meng.

4 • 15:00–15:30 Masayo Fujimura ([email protected]) Department of Mathematics, National Defense Academy of Japan, Japan Title: Geometry of ﬁnite Blaschke products: pentagons and pentagrams Abstract: In this talk, I treat two types of curves induced by Blaschke product. For a Blaschke product B of degree d and λ on the unit circle, let ℓλ be the set of lines joining −1 each distinct two preimages in B (λ). The envelope of the family of lines {ℓλ}λ∈∂D is called the interior curve associated with B. In 2002, Daepp, Gorkin, and Mortini proved the interior curve associated with a Blaschke product of degree 3 forms an ellipse. −1 While let Lλ be the set of lines tangent to the unit circle at the d preimages B (λ) and consider the trace of the intersection points of each two elements in Lλ as λ ranges over the unit circle. This trace is called the exterior curve associated with B. The exterior curve associated with a Blaschke product of degree 3 forms a non-degenerate conic. In this talk, I explain the existence of a duality-like geometrical property lies between the interior curve and the exterior curve. Using this property, I create some examples of Blaschke products whose interior curves consist of two ellipses.

z2−0.47 z2−0.2 Figure 1: The envelope indicates the interior curve of B(z) = z 1−0.47z2 1−0.2z2 . The interior curve consists of two ellipses, one is inscribed in the family of pentagons and the other is inscribed in the family of pentagrams.

5 • 16:00–16:30 Patrick X. Rault ([email protected]) Department of Mathematics, University of Nebraska at Omaha, USA Title: Singularities of Base Polynomials and Gau-Wu Numbers Abstract: In 2013, Gau and Wu introduced a unitary invariant, denoted by k(A), of an n × n matrix A, which counts the maximal number of orthonormal vectors xj such that the scalar products ⟨Axj, xj⟩ lie on the boundary of the numerical range W (A). We refer to k(A) as the Gau–Wu number of the matrix A. We write H1 and iH2 for the Hermitian and skew-Hermitian parts of A (respectively), and use them to define the base polynomial F (x, y, t) = det(xH1 + yH2 + tI). In this talk we will take an algebraic-geometric approach and consider the effect of the singularities of the base curve F (x : y : t) = 0, whose dual is the boundary generating curve, to classify k(A). This continues the work of Wang and Wu classifying the Gau-Wu numbers for 3 × 3 matrices. Our focus on singularities is inspired by Chien and Nakazato, who classified W (A) for 4 × 4 unitarily irreducible A with irreducible base curve according to singularities of that curve. When A is a unitarily irreducible n×n matrix, we give necessary conditions for k(A) = 2, characterize k(A) = n, and apply these results to the case of unitarily irreducible 4 × 4 matrices. Co-author(s): Kristin A. Camenga (Houghton College), Louis Deaett (Quinnipiac Uni- versity), Tsvetanka Sendova (Michigan State University), Ilya M. Spitkovsky (New York University Abu Dhabi), Rebekah B. Johnson Yates (Houghton College)

• 16:30–17:00 Michiya Mori ([email protected]) Graduate School of Mathematical Sciences, the University of Tokyo, Japan Title: Order isomorphisms of von Neumann algebras Abstract: I will consider the usual order structure of self-adjoint parts of von Neumann algebras. I will explain the general form of order isomorphisms between intervals of von Neumann algebras. In particular, I will explain that every order isomorphism between the positive cones of von Neumann algebras without commutative direct summands extends to a linear mapping.

6 • 17:00–17:30 An-Bao Xu ([email protected]) College of Mathematics, Physics and Electronic Engineering, Wenzhou University, China Title: Parametrized quasi-soft thresholding operator for compressed sensing and matrix completion Abstract: Compressed sensing and matrix completion are two new approaches to signal acquisition and processing. Even though the two approaches are diﬀerent, there is a close connection between them. In compressed sensing, based on four basic operator, we give a parametrized quasi-soft thresholding operator and its induced algorithm. Further, by updating parametrized quasi-soft thresholding operator in every iteration, the varied para- metric quasi-soft thresholding algorithm is obtained. Then we generalize both algorithms to suit matrix completion. Finally, the convergence of all algorithms are proved, and the numerical results given show that the new algorithms can eﬀectively improve the accuracy to achieve compressed sensing and matrix completion. Co-author(s): Hugo J. Woerdeman.

7 23, June (Sunday)

• 10:00–10:30 Chi-Kwong Li ([email protected]) Department of Mathematics, College of William and Mary, USA Title: Numerical ranges, operator systems, and quantum channels Abstract: We describe some problems and results on numerical ranges related to operator systems associated with quantum channels.

• 10:30–11:00 NatáliaBebiano ([email protected]) Department of Mathematics, University of Coimbra, Portugal Title: The thermodynamics of systems described by non-Hermitian Hamiltonian operators Abstract: The appearance, in quantum physics, of non-Hermitian Hamiltonians possess- ing a discrete real spectrum inspired a remarkable research activity. In this talk we revisit standard concepts of thermodynamics for systems described by a non-Hemitian Hamil- tonian with real eigenvalues. We mainly focus on the standard case where the energy is the unique conserved quantity. However, other conserved quantities may be considered. Numerical range techniques are used. Co-author: Joãoda Providência.

• 11:15–11:45 Seung-Hyeok Kye ([email protected]) Department of Mathematics, Seoul National University, Seoul, Korea Title: Roles of exposed indecomposable positive multi-linear maps in quantum information theory Abstract: Positive multi-linear maps play essential roles to detect multi-partite entanglement in quantum information theory. We need indecomposable positive maps in order to PPT entanglement. We discuss what kinds of map do we need to detect nonzero volume of PPT entanglement. Exposedness arise naturally in this context. We exhibit several examples of exposed indecomposable positive maps in 3 ⊗ 3 and 2 ⊗ 2 ⊗ 2 cases. We also discuss how these maps can be used to study the structures of the convex set consisting of separable states.

8 • 11:45–12:15 G. Ramesh ([email protected]) Department of Mathematics, I. I. T. Hyderabad, Sangareddy, Telangana, India-502 285 Title: On a subclass of norm attaining operators Abstract: A bounded linear operator T : H → H, where H is a Hilbert space, is said to be norm attaining if there exists a unit vector x ∈ H such that ∥T x∥ = ∥T ∥. Let RT denote the set of all reducing subspaces of T . Deﬁne

β(H) := {T ∈ B(H): T |M : M → M is norm attaining for every M ∈ RT }.

In this talk, we discuss properties and structure of positive operators in β(H) and compare with those of absolutely norm attaining operators (AN -operators). Co-author(s): Hiroyuki Osaka, Ritsumeikan University, BKC Campus, Kusatsu, Japan.

• 14:00–14:30 Yongdo Lim ([email protected]) Department of Mathematics, Sungkyunkwan University, Korea Title: Polar decompositions and Aluthge transforms Abstract: We introduce a new polar decomposition on the Lie group of invertible matrices

M = P tUP, t ≠ 1

1+t and the absolute map |M|t := P (the usual polar decomposition is when t = 0). In this talk, we discuss the corresponding Aluthge transform

△ | |λ | |1−λ t,λ(M) := M t Ut(M) M t

and its numerical range. Co-author: Jorge Antezana.

• 14:30–15:00 Yuki Seo ([email protected]) Department of Mathematics Education, Osaka Kyoiku University, Japan Title: Numerical radius inequalities related to the geometric means of negative power Abstract: The norm inequalities related to the geometric means are discussed by many researchers. We discuss numerical radius inequalities related to the geometric means. Though the operator norm is unitarily invariant one, the numerical radius is not so and unitarily similar. In this talk, we show numerical radius inequalities related to the geometric means of negative power for positive invertible operators.

9 • 15:00–15:30 Shigeru Furuichi ([email protected]) Department of Information Science, College of Humanities and Sciences, Nihon University, Japan Title: Generalization and improvements of numerical radius inequalities Abstract: Let B (H) denote the C∗-algebra of all bounded linear operators on a complex Hilbert space H. For A ∈ B (H), let w (A) and ∥A∥ denote the numerical radius and the usual operator norm of A, respectively. It is well known that 1 ∥A∥ ≤ w (A) ≤ ∥A∥ . 2 An improvement of the above inequality has been given by Kittaneh in 2005. It says that for A ∈ B (H), 1 1 ∥A∗A + AA∗∥ ≤ w2 (A) ≤ ∥A∗A + AA∗∥ . 4 2 In this talk, after reviewing recent results containing such inequalities, by using the properties of convex functions, we improve under a certain condition and generalize these inequalities. Our trial to obtain improvements for the previous results depend on our tools of mathematical inequalities so that such an application to numerical radius inequalities is a new approach. Thus we hope to be useful for researchers in the ﬁeld. Meanwhile we study this topic with geometrically convex functions, we obtained interesting scalar inequalities as a by-product. Co-author: Hamid Reza Moradi, Department of Mathematics, Payame Noor University (PNU), Iran, [email protected]

• 16:00–16:30 Jinchuan Hou ([email protected]) Department of Mathematics, Taiyuan University of Technology, China Title: Preservers of numerical radius on Lie products of self-adjoint operators

Abstract: Let H be a complex Hilbert space with dim H ≥ 3, Bs(H) the Lie algebra of all bounded self-adjoint operators on H, and let F : B(H) → [d, ∞] with d ≥ 0 be a radial unitary similarity invariant function. A structure feature for maps ϕ on Bs(H) satisfying

F (ϕ(A)ϕ(B) − ϕ(B)ϕ(A)) = F (AB − BA)(A, B ∈ Bs(H))

is given. As an application of this result, a characterization of the maps on Bs(H) preserving the numerical radius, the maps preserving the p-norm, the maps preserving the pseudo spectral radius are obtained. Furthermore, complete classiﬁcation of the maps on Bs(H) preserving the numerical range and the maps preserving the pseudo spectrum are also achieved. Co-author(s): Qingsen Xu (TYUT).

10 • 16:30–17:00 Ming-Cheng Tsai ([email protected]) General Education Center, Taipei Uni- versity of Technology, TAIWAN Title: Nonsurjective maps between rectangular matrices preserving disjointness, (zero) triple product or norms

Abstract: Let Mm,n be the space of m × n real or complex rectangular matrices. Two ∗ ∗ matrices A, B ∈ Mm,n are disjoint if A B = 0n and AB = 0m. In this talk, a characterization is given for linear maps Φ : Mm,n → Mr,s sending disjoint matrix pairs to disjoint matrix pairs, i.e., A, B ∈ Mm,n being disjoint ensures that Φ(A), Φ(B) ∈ Mr,s being disjoint. The result is used to characterize nonsurjective linear maps that pre- serve JB∗-triple product, or just zero triple product, on rectangular matrices, deﬁned by { } 1 ∗ ∗ A, B, C = 2 (AB C + CB A). The result is also applied to characterize linear maps between rectangular matrix spaces of diﬀerent sizes preserving the Schatten p-norms or the Ky Fan k-norms. Co-author(s): Chi-Kwong Li, Ya-Shu Wang, Ngai-Ching Wong.

• 17:00–17:30 Shuhei Wada ([email protected]) Department of Information and Computer Engineering, National Institute of Technol- ogy(KOSEN), Kisarazu College, Japan. Title: Ando-Hiai type inequalities for operator means and operator perspectives Abstract: When σ is an operator mean in the sense of Kubo-Ando and A, B > 0 are positive invertible operators, the Ando-Hiai inequality is typically stated as follows:

AσB ≤ I ⇒ ApσBp ≤ I (p ≥ 1),

AσB ≥ I ⇒ ApσBp ≥ I (p ≥ 1).

Since the ﬁrst appearance in the case of weighted operator geometric means, Ando-Hiai type inequalities for operator means have been in active consideration, and have taken an important part in recent developments of multivariable operator means, in particular, of multivariable geometric means. We improve the existing Ando-Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie-Trotter formula and consider the extension problem of operator perspectives to non-invertible positive operators. Co-author(s): Fumio Hiai and Yuki Seo.