ABSTRACTS. Operator Theory and Its Applications

Transcription

1 Volume 16(2017) ABSTRACTS Operator Theory and Its Applications June 26 27, 2017 Seoul National University, Seoul, 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 Position of *-homomorphisms Marie Choda 1 Reducing subspaces of the Dirichlet space via local inverses and Riemann surface Shuaibing Luo 2 Gelfand theory unplugged Robin Harte 3 Wavelets and spectral triples for higher-rank graphs Sooran Kang 4 Operator norm inequality and some positive definite functions Albania Nugraha Imam and Masaru Nagisa 5 Determinant computations for operators of Toeplitz/Hankel type Torsten Ehrhardt 6 C-symmetry of asymmetric truncated Toeplitz operators Marek Ptak 7 Weakly peripherally n-tuple multiplicative maps between function algebras R. Shindo Togashi 8 On complex symmetric operators Chafiq Benhida 9 Aluthge transforms of weighted composition operators in L 2 -spaces Piotr Budzynski 10

3 Weighted shifts associated with composition operators and semi-flows George Exner 11 Subnormality of composition operators over one-circuit directed graphs Zenon Jab loński 12 Looking at Browder and Weyl type theorems for Banach space operators through the holes and the isolated points of the Weyl spectrum B.P. Duggal 13 Markov chains and generalized wavelet multiresolutions Myung-Sin Song 14 Composition operators on Fock s type Hilbert spaces of entire functions Jan Stochel 15 Toral and spherical Aluthge transforms and their common invariant subspaces Jasang Yoon 16 Hyponormal Singular Integral Operators and Fourier Series in L 2 Takanori Yamamoto 17 On characterization of truncated Toeplitz operators by conjugations Kamila Kliś-Garlicka 18 Truncated moment problems and the Hadamard product Seonguk Yoo 19 High order isometric composition operators on l p spaces and infinite graphs with polynomial growth Caixing Gu 20

4 Volume 16(2017), 1 1 Position of *-homomorphisms Marie Choda Osaka Kyoiku University, Japan marie@cc.osaka-kyoiku.ac.jp We show two kinds of results related to positions of *-homomorphisms. One is that from the view point of the concept on operational convexity. The other is that from the view point of entropy where doubly stochastic matrices play a vital role.

5 Volume 16(2017), 2 2 Reducing subspaces of the Dirichlet space via local inverses and Riemann surface Shuaibing Luo Hunan University, P.R. China shuailuo2@126.com Suppose T is a bounded linear operator on a Hilbert space H, if a closed subspace M of H is invariant under both T and T, then M is called a reducing subspace of T on H. Reducing subspaces of M B on the Hardy and Bergman spaces have been studied extensively in the past, where B is a finite Blaschke product. But little is known about the reducing subspaces of M B on the Dirichlet space. There has been some progress in studying this. In this talk, we will use local inverses and Riemann surface to discuss the structure of the reducing subspaces of M B on the Dirichlet space. This is a joint work with Caixing Gu and Jie Xiao.

6 Volume 16(2017), 3 3 Gelfand theory unplugged Robin Harte Trinity College, Ireland rharte@maths.tcd.ie It was Norbert Wiener who observed that whenever a periodic continuous function which never vanishes has an absolutely convergent Fourier series, then so does its reciprocal. Pointwise multiplication generates convolution of their coefficient sequences, with a homomorphism from sequences to functions; according to Wiener, if the function is pointwise invertible then also the sequence is convolution invertible. When Israel Gelfand looked at these sequences he saw for the first time what would come to be known as a ommutative Banach algebra. He went on to extend Wiener s observation from absolutely summable sequences to these Banach algebras, with a completely different and abstract proof. The electricity that powers this Gelfand theory is Zorn s lemma and maximal ideals, together with the Gelfand-Mazur lemma, which says that maximal ideals are always generated by bounded multiplicative linear functionals. The unplugged version bypasses maximal ideals, and proceeds via the superficially more concrete spectral mapping theorem for finite and infinite systems of Banach algebra elements. References [1] Robin Harte, Invertibility and singularity, Dekker (New York), [2] Robin Harte, Spectral mapping theorems - bluffer s guide, Springer Briefs in Mathematics, [3] R.E. Harte, Non-commutative Taylor invertibility, Operators and Matrices (to appear). [4] Vladimir Müller, Spectral theory of linear operators, Birkhäuser Basel, 2007.

7 Volume 16(2017), 4 4 Wavelets and spectral triples for higher-rank graphs Sooran Kang Sungkyunkwan University, Korea soorankang@gmail.com We discuss two ways to construct a spectral triples for a finite higher-rank graph (or k-graph) and how these spectral triples are intimately connected to the wavelet decomposition of the infinite path space of a k-graph which was introduced by Farsi, Gillaspy, Kang and Packer in This is a joint work with C. Farsi, E. Gillaspy, A. Julien and J. Packer.

8 Volume 16(2017), 5 5 Operator norm inequality and some positive definite functions Albania Nugraha Imam and Masaru Nagisa Universitas Pendidikan Indonesia, Indonesia; Chiba University, Japen phantasion@gmail.com, nagisa@math.s.chiba-u.ac.jp Hiai-Kosaki considered some order ( ) for functions f α (t) = α 1 α t α 1 t α 1 1 (α R) on [0, ) and proved f α f β if α β. Here, f g means the function R x f(ex ) g(e x ) is positive definite and it is known that f 1/2 f 2 implies McIntosh inequality H 1/2 XK 1/2 1 HX + XK 2 for H, K, X M n (C), H, K 0 and means any unitarily invariant norm. We extend the class of above functions to the following: f α,β (t) = t γ(α,β) k i=1 b i (t a i 1) a i (t b i 1) on [0, ), where α = (a 1, a 2,..., a k ), β = (b 1, b 2,..., b k ) R k and γ(α, β) = 1 k i=1 (a i b i ) 2. We will prove f α,β f α,β for some pairs of α, β Rk and α, β R l.

9 Volume 16(2017), 6 6 Determinant computations for operators of Toeplitz/Hankel type Torsten Ehrhardt University of California, Santa Cruz, U.S.A. tehrhard@ucsc.edu In my talk I will consider two problems that are related to the computation of certain operators determinants. The first problem concerns the asymptotics of the operator determinant of identity plus a Hankel-like operator. More precisely, the latter operators is an integral operator defined on L 2 [ R, ) with kernel h(x + y) and we are interested in the asymptotics R. Here h is the Fourier transform a symbol. None of the plenty results known for finite truncations of Wiener-Hopf plus Hankel operators can be applied directly. Nonetheless, for a well-behaved symbol a kind of Szegö-Achiezer-Kac type formula can be proved. The second problem concerns the exact computation for the constant term in the Szegö-Widom Limit Theorem for block Toeplitz determinants. This constant is an operator determinant E[a] = det T (a)t (a 1 ), which involves the (block) Toeplitz operators with scalar or matrix valued symbols. Only in the scalar case, an explicit formula for E[a] is known, while in the matrix case a general formula is elusive. Nonetheless, for (very) special classes of matrix symbols, the constant can be identified. Both problems arose from concrete applications.

10 Volume 16(2017), 7 7 C-symmetry of asymmetric truncated Toeplitz operators Marek Ptak University of Agriculture in Kraków, Poland rmptak@cyf-kr.edu.pl Let H 2 be the Hardy space on the unit disc, identified as usual with a subspace of L 2 on the unit circle. With any nonconstant inner function θ we associate the model space K 2 θ, defined by K2 θ = H2 θh 2. In this space we can define the conjugation (antilinear, isometric, involution) C θ : K 2 θ K2 θ by C θf(z) = θzf(z). Let us consider two nonconstant inner functions α and θ such that α divides θ. For certain functions ϕ L 2 we can define an asymmetric truncated Toeplitz operator A ϕ : K 2 θ K2 α by A ϕ f = P α (ϕf), where P α : L 2 K 2 α is the orthogonal projection. In a symmetric case, θ = α, bounded truncated Toeplitz operators are C symmetric, i.e. C θ A ϕ C θ = A ϕ. The relation between bounded asymmetric truncated Toeplitz operators with L 2 symbols and conjugations C θ, C α, in an asymmetric case α divides θ, will be shown. Joint work with C. Câmara, K. Kliś Garlicka.

11 Volume 16(2017), 8 8 Weakly peripherally n-tuple multiplicative maps between function algebras R. Shindo Togashi National Institute of Technology, Nagaoka College, Japen rumi@nagaoka-ct.ac.jp We introduce weakly peripherally-multiplicative surjections between function algebras and the following is proved: if the Choquet boundary Ch(B) is first-countable and T : A B is a surjection such that, for a fixed natural number n 2, σ π ( n k=1 T (f k)) σ π ( n k=1 f k) for all f 1,, f n A, where σ π (f) is the peripheral spectrum of f, then there exist a homeomorphism φ : Y X and a continuous function ω : Ch(B) {z : z = 1} such that T (f) = ω (f φ) on Ch(B).

12 Volume 16(2017), 9 9 On complex symmetric operators Chafiq Benhida Université de Lille I, France chafiq.benhida@math.univ-lille1.fr Complex symmetric operators on a Hilbert space are connected to many topics in mathematics and physics and are interesting in many aspects. They have known a great development in the last years. In this talk, we ll discuss among other things their spectral properties.

13 Volume 16(2017), Aluthge transforms of weighted composition operators in L 2 -spaces Piotr Budzynski University of Agriculture in Kraków, Poland piotr.budzynski@ur.krakow.pl The talk is aimed at presenting recent results concerning Aluthge transforms of (unbounded) weighted composition operators acting in L 2 -spaces. Recall that, given a σ-finite measure space (X, A, µ), an A -measurable transformation φ of X and a complex A -measurable function w on X, the weighted composition operator in L 2 (µ) induced by φ and w is given by D(C φ,w ) = {f L 2 (µ): w (f φ) L 2 (µ)}, C φ,w f = w (f φ), f D(C φ,w ), We will show that the α-aluthge transform α (C φ,w ) of a densely defined weighted composition operators C φ,w is a closable operator whose closure is a weighted composition operator C φ,wα induced by φ and a weight function w α that can be written in terms of the transformation φ, the weight function w, and the Radon-Nikodym derivative h φ,w, canonically attached to C φ,w. We will supply conditions for the equality α (C φ,w ) = C φ,wα. We will provide a characteriaztion for p-hyponormality of unbounded weighted composition operators in L 2 -spaces and presents results concerning p-hyponormality of Aluthge transforms of weighted composition operators. The talk is based on a joint work with C. Benhida, J. Stochel, and J. Trepkowski.

14 Volume 16(2017), Weighted shifts associated with composition operators and semi-flows George Exner Bucknell University, U.S.A. Consider a linear fractional transformation ϕ mapping the open unit disk into itself (having 1 as a fixed point) and the associated composition operator C ϕ acting on the Hardy space. The subspace consisting of the span of reproducing kernels corresponding to iterates under ϕ of 0 yields a space on which C ϕ is similar to a weighted shift. We show that the hyponormality and subnormality of this shift are neatly characterized in terms of the location of the other fixed point of ϕ. We consider similar questions for semi-groups of composition operators (semi-flows).

15 Volume 16(2017), Subnormality of composition operators over one-circuit directed graphs Zenon Jab loński Jagiellonian University, Poland We will show that there exists an example of a non-hyponormal injective composition operator in an L 2 -space over a locally finite directed graph that generates Stieltjes moment sequences. The question of how simple such a locally finite directed graph can be will be discussed. The talk is based on joint work with P. Budzyński, I.B. Jung and J. Stochel.

16 Volume 16(2017), Looking at Browder and Weyl type theorems for Banach space operators through the holes and the isolated points of the Weyl spectrum B.P. Duggal University of Nis, Serbia Given a Banach space operator A, let η σ w (A) denote the union of the holes of σ w (A). Then A satisfies Browder s theorem, A (Bt), if and only if A has SVEP on σ(a) η σ w (A) and, letting σ aw (A) denote the upper Weyl spectrum of A, A satisfies a-browder s theorem (A (a Bt)) if and only if A has SVEP on σ a (A) η σ aw (A). Again, if we let E 0 (A) (E0 a (A)) denote the set of finite multiplicity isolated eigenvalues of A (the set of finite multiplicity eigenvalues in ıσ a (A)), then A satisfies Weyl s theorem, A (W t), if and only if A (Bt) and E 0 (A) σ w (A)} = and A (a W t) if and only if A (a Bt) and E0 a(a) σ aw(a)} =. Similar assertions hold for the generalized versions, i.e. the B-Browder and B-Weyl versions [1], of these results. Browder s theorem type results survive perturbation by commuting Riesz operators R, but this does not extend to Weyl type theorems. A typical requirement here for the transfer of (W t) and a W t (or their generalized versions) from A to A + R is that A is finitely isoloid and (respectively) A is finitely a-isoloid. As an immediate consequence of these observations one sees that the perturbation by a commuting Riesz operator of: (i) Analytic Toeplitz operators T f (σ(t f ) = σ w (T f ) is connected) and non-quasinilpotent operators A B(l p ), 1 p <,) satisfying the abstract shift condition (σ(a) = σ w (A) is connected) satisfy (W t) and generalised (W t); (ii) weighted right shift operators A (σ a (A) = σ aw (A) is connected) satisfy (a W t) and generalised (a W t). Totally hereditarily normaloid (in particular, paranormal) operators and subscalar operators (more generally operators A for which the quasinilpotent part H 0 (A λ) = (A λ) p (0), all complex λ, for some integer p 0) are polaroid and have SVEP, hence satisfy (W t) and generalised (W t). References [1] M. Berkani and J. J. Koliha, Weyl type theorems for bounded linear operators, Acta Math. Sci.(Szeged) 69(2003), [2] B. P. Duggal, Spectral picture, perturbed Browder and Weyl theorems, and their variations, Functional Analysis Appoximation and Computation 9(1)(2017), 1 23.

17 Volume 16(2017), Markov chains and generalized wavelet multiresolutions Myung-Sin Song Southern Illinois University, U.S.A. We show how some orthonormal bases can be generated by representations of the Cuntz algebra with Markov chains.

18 Volume 16(2017), Composition operators on Fock s type Hilbert spaces of entire functions Jan Stochel Jagiellonian University, Poland Jan.Stochel@im.uj.edu.pl Composition operators with analytic symbols on Fock s type reproducing kernel Hilbert spaces will be discussed. These spaces are build over complex Hilbert spaces and they are induced by entire functions with non-negative Taylor s coefficients. We will pay particular attention to the issue of boundedness. The case of composition operators on the Segal- Bargmann space of finite and infinite order will be discussed as well.

19 Volume 16(2017), Toral and spherical Aluthge transforms and their common invariant subspaces Jasang Yoon University of Texas Rio Grande Valley, U.S.A. Recently, R. Curto and J. Yoon have introduced the toral and spherical Aluthge transforms for commuting pairs (with particular emphasis on spherically quasinormal and spherically isometric 2-variable weighted shifts) and studied their basic properties. In this talk, we introduce and investigate nontrivial common invariant subspaces between the toral (resp. spherical) Aluthge transform and the original n-tuple of bounded operators with dense ranges. This is a joint work with Jaewoong Kim.

20 Volume 16(2017), Hyponormal Singular Integral Operators and Fourier Series in L 2 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). This paper is concerned with the 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). In this paper, we study the hyponormality of S α,β which is related to the hyponormal Toeplitz operator on H 2 (T). We consider the condition of Fourier series of α and β such that S α,β is hyponormal.

21 Volume 16(2017), On characterization of truncated Toeplitz operators by conjugations Kamila Kliś-Garlicka University of Agriculture in Kraków, Poland Let θ be a nonconstant inner function. Denote by K θ the so called model space given by K θ = H 2 θh 2. For a function ϕ L 2 define a truncated Toeplitz operator A ϕ : K θ K θ, A ϕ f = P θ (ϕf), where P θ : L 2 K θ is the orthogonal projection. Truncated Toeplitz operators are C symmetric with respect to the canonical conjugation given on an appropriate model space. However, by considering only one conjugation one cannot characterize truncated Toeplitz operators. It will be proved that if an operator on a model space is C symmetric for a certain family of conjugations in the model space, then is has to be a truncated Toeplitz operator. A characterization of classical Toeplitz operators is also presented in terms of conjugations. The talk is based on common work with Bartosz Lanucha and Marek Ptak.

22 Volume 16(2017), Truncated moment problems and the Hadamard product Seonguk Yoo Sungkyunkwan University, Korea The best solution to the truncated moment problem (TMP) for now is probably the Flat Extension Theorem, which states if the moment matrix of a moment sequence admits a rank-preserving positive extension, then the sequence has a representing measure. We consequently need to build a positive moment matrix extension. However, construction of a flat extension is not easy for most higher-order moment sequences since we need to allow many parameters for an extension. As a new approach, the author recently has considered various decompositions of a moment matrix to seek a solution to TMP instead of an extension. It is well-known that column relations in the moment matrix store critical information about solutions to TMP. We naturally sense that the more relations bring more information and make the problem easier. On the contrary, if the moment matrix has one or no column relation, the corresponding problems are very difficult. Using the rank-one decomposition of a positive matrix and the Hadamard product, we would like to solve TMP with a certain single column relation.

23 Volume 16(2017), High order isometric composition operators on l p spaces and infinite graphs with polynomial growth Caixing Gu California Polytechnic State University, U.S.A. cgu@calpoly.edu Let ϕ be a map of natural numbers to natural numbers. We characterize composition operators C ϕ on l p that are (m, q)-isometries. We observe that C ϕ on l p is an (m, p)- isometry for one particular p 1 if and only if C ϕ on l p is an (m, p)-isometry for all p 1. We then discuss C ϕ on l 2. We prove that for an m-isometric C ϕ, the covariance operator β m 1 (C ϕ ) is a finite rank operator if and only if ϕ has a periodic point. We completely classify, up to unitary equivalence, all C ϕ that are 2-isometries. Numerous examples of m-isometric C ϕ for all m 2 are constructed using rooted (periodic) or unrooted (aperiodic) infinite graphs having polynomial growth. We also prove that if C ϕ on l p is an (m, q)-isometry, then q = np for some integer n.