D. Luecking’s Finite Rank Theorem for Toeplitz Operator, Benedicks’ Theorem on the Heisenberg Group, and Uncertainty Principle for the Fourier–Wigner Transform

2018 ◽  
Vol 235 (2) ◽  
pp. 208-219
Author(s):  
A. Samanta ◽  
S. Sarkar
Keyword(s):  
Heisenberg Group ◽  
Toeplitz Operator ◽  
Uncertainty Principle ◽  
Finite Rank ◽  
Wigner Transform ◽  
Rank Theorem

scholarly journals The Detectable Subspace for the Friedrichs Model: Applications of Toeplitz Operators and the Riesz–Nevanlinna Factorisation Theorem

2020 ◽  
Vol 21 (10) ◽  
pp. 3141-3156
Author(s):  
S. Naboko ◽  
I. Wood
Keyword(s):  
Toeplitz Operator ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Model Operator ◽  
Hardy Classes ◽  
Finite Rank Perturbation ◽  
Boundary Measurements ◽  
Independent Variable ◽  
Friedrichs Model

Abstract We discuss how much information on a Friedrichs model operator (a finite rank perturbation of the operator of multiplication by the independent variable) can be detected from ‘measurements on the boundary’. The framework of boundary triples is used to introduce the generalised Titchmarsh–Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. In this paper, we choose functions arising as parameters in the Friedrichs model in certain Hardy classes. This allows us to determine the detectable subspace by using the canonical Riesz–Nevanlinna factorisation of the symbol of a related Toeplitz operator.

scholarly journals THE FINITE SUM OF THE PRODUCTS OF TWO TOEPLITZ OPERATORS

2009 ◽  
Vol 86 (1) ◽  
pp. 45-60 ◽  
Author(s):  
XUANHAO DING
Keyword(s):  
Toeplitz Operator ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rank Perturbation ◽  
Sum Of Products ◽  
Finite Sum ◽  
Necessary And Sufficient

AbstractWe consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator.

HARDY AND MIYACHI THEOREMS FOR HEISENBERG MOTION GROUPS

2016 ◽  
Vol 229 ◽  
pp. 1-20 ◽  
Author(s):  
ALI BAKLOUTI ◽  
SUNDARAM THANGAVELU
Keyword(s):  
Heisenberg Group ◽  
Uncertainty Principle ◽  
Price Uncertainty ◽  
Motion Group ◽  
Hardy’S Theorem ◽  
Motion Groups ◽  
Miyachi’S Theorem

Let $G=\mathbb{H}^{n}\rtimes K$ be the Heisenberg motion group, where $K=U(n)$ acts on the Heisenberg group $\mathbb{H}^{n}=\mathbb{C}^{n}\times \mathbb{R}$ by automorphisms. We formulate and prove two analogues of Hardy’s theorem on $G$. An analogue of Miyachi’s theorem for $G$ is also formulated and proved. This allows us to generalize and prove an analogue of the Cowling–Price uncertainty principle and prove the sharpness of the constant $1/4$ in all the settings.

scholarly journals Algebraic Properties of Toeplitz Operators on the Pluriharmonic Bergman Space

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jingyu Yang ◽  
Liu Liu ◽  
Yufeng Lu
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Rank Product ◽  
Algebraic Properties ◽  
Necessary And Sufficient ◽  
Product Problem ◽  
Quasihomogeneous Symbols

We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space. We first give the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator and discuss the zero-product problem for several Toeplitz operators with radial symbols. Next, we study the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols. Finally, we also investigate finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols.

scholarly journals The Finite Rank Theorem for Toeplitz Operators on the Fock Space

2013 ◽  
Vol 25 (1) ◽  
pp. 347-356 ◽  
Author(s):  
Alexander Borichev ◽  
Grigori Rozenblum
Keyword(s):  
Finite Rank ◽  
Toeplitz Operators ◽  
Fock Space ◽  
Rank Theorem

scholarly journals Valuated Butler groups of finite rank

2006 ◽  
Vol 80 (3) ◽  
pp. 335-350
Author(s):  
L. Fuchs ◽  
K. M. Rangaswamy
Keyword(s):  
Finite Rank ◽  
Special Kind ◽  
Full Characterization ◽  
Rank Theorem

AbstractValuated Butler groups of finite rank are investigated. The valuated B2-groups are both epic images and pure subgroups of completely decomposable valuated groups of finite rank (Theorem 3.1). However, there are fundamental changes in the theory of Butler groups when valuations are involved. We introduce valuated B1-groups and show that they are valuated B2-groups. Surprisingly, valuated B2-groups of rank greater than 1 need not be valuated B1 -groups, unless they carry a special kind valuation, see Theorem 7.1. Theorem 6.5 gives a full characterization of valuated B1 -groups of finite rank, generalizing Bican's characterization of Butler groups.

scholarly journals Uncertainty Principles for the Dunkl-Wigner Transforms

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Fethi Soltani
Keyword(s):  
Uncertainty Principle ◽  
Uncertainty Principles ◽  
Wigner Transform ◽  
Heisenberg Type ◽  
Local Uncertainty Principle ◽  
Local Uncertainty

We prove a version of Heisenberg-type uncertainty principle for the Dunkl-Wigner transform of magnitude s>0; and we deduce a local uncertainty principle for this transform.

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