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About 1,073 papers for “Toeplitz operator”
2015
Abstract. We show that, perhaps surprisingly, the asymptotic behaviour of the Berezin transform as well as some properties of Toeplitz operators on a variety of weighted harmonic and pluriharmonic Bergman spaces seem to be the same as in the holomorphic case. Let Ω be a bounded domain in Cn, L2hol(Ω) ⊂ L2(Ω) the Bergman space of all square-integrable holomorphic functions on Ω, and K(x, y) its reproducing kernel, i.e. the Bergman kernel. Thus f(x) = Ω
Cited by 0
Qian Ding, Yong Chen · AIMS Mathematics · 2023
<abstract><p>In this paper, we characterize when the product of two H-Toeplitz operators to be another H-Toeplitz with one general and another quasihomogeneous symbols. Also, we describe the product of H-Toeplitz operator and Toeplitz operator to be another H-Toeplitz with certain harmonic symbols.</p></abstract>
Cited by 4Open access
璎元 曹 · Pure Mathematics · 2025
本文研究了调和Bergman空间中的H-Toeplitz算子，并定义了调和Bergman空间上的JH-Toeplitz算子，给出了两个以拟齐次函数为符号的H-Toeplitz算子的交换性的充要条件。本文还讨论了以拟齐次函数为符号的H-Toeplitz算子与JH-Toeplitz算子乘积的相关理论。This article studies the H-Toeplitz operator in harmonic Bergman space and defines the JH-Toeplitz operator on harmonic Bergman space. The necessary and sufficient conditions for the commutativity of two H-Toeplitz operators with quasi homogeneous functions as symbols are given. The article also discusses the relevant theories of the product of H-Toeplitz operator and JH-Toeplitz operator with quasi homogeneous functions as symbols.
Cited by 0Open access
Espoo A, Jarmo Malinen, Teknillinen Korkeakoulu et al. · 1999
Cited by 0
Douglas Farenick · Transactions of the American Mathematical Society Series B · 2021
A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Toeplitz matrices into the algebra of all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi Subscript n"> <mml:semantics> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\xi _n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generates an extremal ray in the cone of all continuous <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Toeplitz-matrix valued functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the unit circle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">S^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose Fourier coefficients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove f With caret left-parenthesis k right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>f</mml:mi> <mml:mo stretchy="false">^</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\hat f(k)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vanish for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue k EndAbsoluteValue greater-than-or-equal-to n"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≥</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">|k|\geq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Lastly, it is noted that all positive Toeplitz matrices over nuclear C<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi/> <mml:mo>∗</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras are approximately separable.
Cited by 3Open access
O. B. Chernobai · Birkhäuser Basel eBooks · 2009
In this report, we sketch the proof of an integral representation for operator-valued Toeplitz kernels. The proof is based on the spectral theory for the corresponding differential operator constructed from this kernel and acting in Hilbert space. The report also contains references to other new results concerning such Toeplitz kernels.
Cited by 0
金金 梁 · Advances in Applied Mathematics · 2021
Cited by 1
Woo Young Lee · 2001
In this article we give open problems concerning algebraic properties, subnormality, hyponormality, and spectral properties of Toeplitz operators.
Cited by 0
Carl C. Cowen · Journal of Functional Analysis · 1980
Cited by 43
Stephan Ramon Garcia, William T. Ross · CRM proceedings & lecture notes · 2010
Abstract. We prove several lower bounds for the norm of a truncated Toeplitz operator and obtain a curious relationship between the H 2 and H ∞ norms of functions in model spaces. 1.
Cited by 9
Carl C. Cowen · Transactions of the American Mathematical Society · 1978
For a function <italic>f</italic> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit disk, the operator on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of multiplication by <italic>f</italic> will be denoted by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript f"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_f}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and its commutant by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript f Baseline right-brace prime"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {T_f}\} ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For a finite Blaschke product <italic>B</italic>, a representation of an operator in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript upper B Baseline right-brace prime"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\{ {T_B}\}’}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a function on the Riemann surface of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript negative 1 Baseline ring upper B"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>∘</mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{B^{ - 1}} \circ B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> motivates work on more general functions. A theorem is proved which gives conditions on a family <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> functions which imply that there is a function <italic>h</italic> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript h Baseline right-brace prime equals intersection left-brace upper T Subscript f Baseline right-brace prime"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo>∩</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {T_h}\} ’ = { \cap _{f \in \mathcal {F}}}\{ {T_f}\} ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a special case of this theorem, we find that if the inner factor of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f minus f left-parenthesis c right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>−</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>c</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f - f(c)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a finite Blaschke product for some <italic>c</italic> in the disk, then there is a finite Blaschke product <italic>B</italic> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript f Baseline right-brace prime equals left-brace upper T Subscript upper B Baseline right-brace prime"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> <mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ {T_f}\} ’ = \{ {T_B}\} ’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Necessary and sufficient conditions are given for an operator to commute with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript f"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{T_f}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <italic>f</italic> is a covering map (in the sense of Riemann surfaces). If <italic>f</italic> and <italic>g</italic> are in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f equals h ring g"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mi>h</mml:mi> <mml:mo>∘</mml:mo> <mml:mi>g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f = h \circ g</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript f Baseline right-brace prime superset-of left-brace upper T Subscript g Baseline right-brace prime"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>T</mml:mi>
Cited by 115Open access
Agapitos Hatzinikitas · Journal of Mathematical Physics · 2009
Confining a quantum particle in a compact subinterval of the real line with Dirichlet boundary conditions, we identify the connection of the one-dimensional fractional Schödinger operator with the truncated Toeplitz matrices. We determine the asymptotic behavior of the product of eigenvalues for the α-stable symmetric laws by employing the Szegö’s strong limit theorem. The results of the present work can be applied to a recently proposed model for a particle hopping on a bounded interval in one dimension whose hopping probability is given a discrete representation of the fractional Laplacian.
Cited by 1Open access
Joel H. Shapiro · Journal of Mathematical Analysis and Applications · 2006
Cited by 11
E. M. Klein · Proceedings of the American Mathematical Society · 1972
In this paper we explicitly compute the numerical range of an arbitrary Toeplitz operator on the classical Hardy space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the unit circle. In particular, we show that the numerical range depends only on the spectrum of the given Toeplitz operator. Several special cases are also considered.
Cited by 29
Hatzinikitas, Agapitos · arXiv (Cornell University) · 2009
Confining a quantum particle in a compact subinterval of the real line with Dirichlet boundary conditions, we identify the connection of the one-dimensional fractional Schrödinger operator with the truncated Toeplitz matrices. We determine the asymptotic behaviour of the product of eigenvalues for the $α$-stable symmetric laws by employing the Szegö's strong limit theorem. The results of the present work can be applied to a recently proposed model for a particle hopping on a bounded interval in one dimension whose hopping probability is given a discrete representation of the fractional Laplacian.
Cited by 0Open access
Keras, S, Cambridge Univ. (United Kingdom). Dept. of Applied Mathematics and Theoretical Physics · OpenGrey (Institut de l'Information Scientifique et Technique) · 1995
Available from British Library Document Supply Centre- DSC:9106.1605(CU-DAMTP-NA--04/1995) / BLDSC - British Library Document Supply Centre
Cited by 0
Bauer, Wolfram 1974-, Duduchava, Roland 1945-, Grudskij, Sergej M. 1955- et al.
Cited by 0
Lan Zhou · 2011
Cited by 0
Saifallah Ghobber, Hatem Mejjaolı · Journal of Pseudo-Differential Operators and Applications · 2025
Cited by 0
Yinyin Hu, Jia Deng, Yu Tao et al. · Journal of Function Spaces · 2018
We define the dual truncated Toeplitz operators and give some basic properties of them. In particular, spectrum and reducing subspaces of some special dual truncated Toeplitz operator are characterized.
Cited by 6Open access