scholarly journals Restriction of Toeplitz Operators on Their Reducing Subspaces

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Anjian Xu ◽  
Yang Zou
Keyword(s):  
Toeplitz Operator ◽  
Unit Disc ◽  
Toeplitz Operators ◽  
Reducing Subspaces ◽  
Bergman Shift ◽  
Analytic Toeplitz Operator

We study the restrictions of analytic Toeplitz operator on its minimal reducing subspaces for the unit disc and construct their models on slit domains. Furthermore, it is shown that Tzn is similar to the sum of n copies of the Bergman shift.

On Quasisimilarity for Analytic Toeplitz Operators

1988 ◽  
Vol 31 (1) ◽  
pp. 111-116 ◽  
Author(s):  
Katsutoshi Takahashi
Keyword(s):  
Toeplitz Operator ◽  
Blaschke Product ◽  
Toeplitz Operators ◽  
Finite Blaschke Product ◽  
Analytic Toeplitz Operator

AbstractLet f be a function in H∞. We show that if f is inner or if the commutant of the analytic Toeplitz operator Tf is equal to that of Tb for some finite Blaschke product b, then any analytic Toeplitz operator quasisimilar to Tf is unitarily equivalent to Tf.

scholarly journals Range Spaces of Co-Analytic Toeplitz Operators

2018 ◽  
Vol 70 (6) ◽  
pp. 1261-1283 ◽  
Author(s):  
Emmanuel Fricain ◽  
Andreas Hartmann ◽  
William T. Ross
Keyword(s):  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Boundary Behavior ◽  
Model Space ◽  
Orthogonal Complement ◽  
Dirichlet Spaces ◽  
Backward Shift ◽  
Weighted Dirichlet Spaces ◽  
General Abstract ◽  
Analytic Toeplitz Operator

AbstractIn this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

scholarly journals Reducing Subspaces of the Dual Truncated Toeplitz Operator

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yinyin Hu ◽  
Jia Deng ◽  
Tao Yu ◽  
Liu Liu ◽  
Yufeng Lu
Keyword(s):  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Truncated Toeplitz Operator ◽  
Basic Properties ◽  
Reducing Subspaces

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.

scholarly journals The spectra of Toeplitz operators with unimodular symbols

1998 ◽  
Vol 41 (1) ◽  
pp. 133-139 ◽  
Author(s):  
Takahiko Nakazi
Keyword(s):  
Toeplitz Operator ◽  
Unit Disc ◽  
Toeplitz Operators ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disc

The spectrum σ(Tφ) of a Toeplitz operator Tφ on the open unit disc D for a unimodular symbol φ is studied and many sufficient conditions for σ(Tφ)⊆∂D or σ(Tφ) = are given. In particular if φ is a unimodular function in H∞ + C, then σ(Tφ)⊆∂D or σ(Tφ) =

scholarly journals A Note on Reducing Subspaces of Toeplitz Operator on the Weighted Analytic Function Spaces of the Bidisk Hw2D2

2017 ◽  
Vol 2017 ◽  
pp. 1-4
Author(s):  
Hongzhao Lin
Keyword(s):  
Analytic Function ◽  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Function Spaces ◽  
Type I ◽  
Reducing Subspaces ◽  
Analytic Function Spaces

We discussed the reducing subspaces of Toeplitz operators Tz1Nz2M  N≠M on the weighted analytic function spaces of bidisk Hw2(D2). The result shows that if the weight w is of type-I, the structure of reducing subspaces of Tz1Nz2M  N≠M on Hw2D2 is very simple.

scholarly journals Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sumin Kim ◽  
Jongrak Lee
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Toeplitz Operators ◽  
Bergman Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

AbstractIn this paper, we present some necessary and sufficient conditions for the hyponormality of Toeplitz operator $T_{\varphi }$ T φ on the Bergman space $A^{2}(\mathbb{D})$ A 2 ( D ) with non-harmonic symbols under certain assumptions.

scholarly journals Toeplitz Operators on Dirichlet-Type Space of Unit Ball

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Jin Xia ◽  
Xiaofeng Wang ◽  
Guangfu Cao
Keyword(s):  
Toeplitz Operator ◽  
Boundary Point ◽  
Toeplitz Operators ◽  
Radial Function ◽  
Type Space ◽  
Dirichlet Type ◽  
Algebraic Properties ◽  
Commuting Problem ◽  
Product Problem ◽  
Dirichlet Type Space

We construct a functionuinL2Bn, dVwhich is unbounded on any neighborhood of each boundary point ofBnsuch that Toeplitz operatorTuis a Schattenp-class0<p<∞operator on Dirichlet-type spaceDBn, dV. Then, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Dirichlet-type spaceDBn, dV. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. We investigate the zero-product problem for several Toeplitz operators with radial symbols. Furthermore, the corresponding commuting problem of Toeplitz operators whose symbols are of the formξkuis studied, wherek ∈ Zn,ξ ∈ ∂Bn, anduis a radial function.

scholarly journals Hyponormal Toeplitz operators on H2(T) with polynomial symbols

1996 ◽  
Vol 144 ◽  
pp. 179-182 ◽  
Author(s):  
Dahai Yu
Keyword(s):  
Functional Equation ◽  
Hardy Space ◽  
Closed Form ◽  
Unit Circle ◽  
Toeplitz Operator ◽  
Explicit Expression ◽  
Complex Plane ◽  
Toeplitz Operators ◽  
Computing Process

Let T be the unit circle on the complex plane, H2(T) be the usual Hardy space on T, Tø be the Toeplitz operator with symbol Cowen showed that if f1 and f2 are functions in H such that is in Lø, then Tf is hyponormal if and only if for some constant c and some function g in H∞ with Using it, T. Nakazi and K. Takahashi showed that the symbol of hyponormal Toeplitz operator Tø satisfies and and they described the ø solving the functional equation above. Both of their conditions are hard to check, T. Nakazi and K. Takahashi remarked that even “the question about polynomials is still open” [2]. Kehe Zhu gave a computing process by way of Schur’s functions so that we can determine any given polynomial ø such that Tø is hyponormal [3]. Since no closed-form for the general Schur’s function is known, it is still valuable to find an explicit expression for the condition of a polynomial á such that Tø is hyponormal and depends only on the coefficients of ø, here we have one, it is elementary and relatively easy to check. We begin with the most general case and the following Lemma is essential.

scholarly journals Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols

2011 ◽  
Vol 2011 ◽  
pp. 1-15
Author(s):  
Zhi Ling Sun ◽  
Yu Feng Lu
Keyword(s):  
Toeplitz Operator ◽  
Bergman Space ◽  
Spectral Decomposition ◽  
Toeplitz Operators ◽  
Complete Information ◽  
Isometric Isomorphism

We construct an operatorRwhose restriction onto weighted pluriharmonic Bergman Spacebμ2(Bn)is an isometric isomorphism betweenbμ2(Bn)andl2#. Furthermore, using the operatorRwe prove that each Toeplitz operatorTawith radial symbols is unitary to the multication operatorγa,μI. Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about the operator, providing its spectral decomposition.

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