scholarly journals Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

2020 ◽  
Vol 76 (1) ◽  
Author(s):  
M. C. Câmara ◽  
M. T. Malheiro ◽  
J. R. Partington
Keyword(s):  
Unit Circle ◽  
Hardy Spaces ◽  
Toeplitz Operators ◽  
Integer Power ◽  
Piecewise Continuous ◽  
Toeplitz Kernel

AbstractWe consider kernels of unbounded Toeplitz operators in $$H^p({\mathbb {C}}^{+})$$ H p ( C + ) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in $$H^p({\mathbb {C}}^{+})$$ H p ( C + ) , we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of z. We apply the results to describe the kernels of Toeplitz operators with non-vanishing piecewise continuous symbols.

Hankel operators with PC symbols and the space H∞ + PC

1981 ◽  
Vol 89 (1-2) ◽  
pp. 17-24 ◽  
Author(s):  
F. F. Bonsall ◽  
T. A. Gillespie
Keyword(s):  
Unit Circle ◽  
Explicit Formula ◽  
Hardy Spaces ◽  
Hankel Operator ◽  
Essential Norm ◽  
Hankel Operators ◽  
Continuous Functions ◽  
Piecewise Continuous ◽  
The Way

SynopsisWe obtain an explicit formula for the essential norm of a Hankel operator with its symbol in the space PC, which is the closure in L∞ of the space of piecewise continuous functions on the unit circle . It follows from this formula that functions in PC can be approximated as closely by functions in C, the continuous functions on the circle, as by functions in the much larger space H∞ + C. This is an example of the way in which properties of the Hardy spaces can be derived from properties of Hankel operators.

scholarly journals Toeplitz operators with PQC symbols on weighted Hardy spaces

1991 ◽  
Vol 97 (1) ◽  
pp. 194-214 ◽  
Author(s):  
Albrecht Böttcher ◽  
Ilya M Spitkovsky
Keyword(s):  
Hardy Spaces ◽  
Toeplitz Operators ◽  
Weighted Hardy Spaces

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 generalized Bergman-Hardy spaces

2001 ◽  
Vol 88 (1) ◽  
pp. 96
Author(s):  
Wolfgang Lusky
Keyword(s):  
Hardy Spaces ◽  
Toeplitz Operators ◽  
Fock Space ◽  
Bergman Spaces ◽  
Trigonometric Polynomials ◽  
Essential Spectra ◽  
Radial Functions ◽  
Hardy And Bergman Spaces ◽  
Schmidt Operator ◽  
Vector Valued

We study the Toeplitz operators $T_f: H_2 \to H_2$, for $f \in L_\infty$, on a class of spaces $H_2$ which in- cludes, among many other examples, the Hardy and Bergman spaces as well as the Fock space. We investigate the space $X$ of those elements $f \in L_\infty$ with $\lim_j \|T_f-T_{f_j}\|=0$ where $(f_j)$ is a sequence of vector-valued trigonometric polynomials whose coefficients are radial functions. For these $T_f$ we obtain explicit descriptions of their essential spectra. Moreover, we show that $f \in X$, whenever $T_f$ is compact, and characterize these functions in a simple and straightforward way. Finally, we determine those $f \in L_\infty$ where $T_f$ is a Hilbert-Schmidt operator.

scholarly journals A Toeplitz-Like Operator with Rational Symbol Having Poles on the Unit Circle III: The Adjoint

2019 ◽  
Vol 91 (5) ◽  
Author(s):  
G. J. Groenewald ◽  
S. ter Horst ◽  
J. Jaftha ◽  
A. C. M. Ran
Keyword(s):  
Unit Circle ◽  
Toeplitz Operator ◽  
Toeplitz Operators ◽  
Adjoint Operator ◽  
Selfadjoint Extension ◽  
Densely Defined

Abstract This paper contains a further analysis of the Toeplitz-like operators $$T_\omega $$ T ω on $$H^p$$ H p with rational symbol $$\omega $$ ω having poles on the unit circle that were previously studied in Groenewald (Oper Theory Adv Appl 271:239–268, 2018; Oper Theory Adv Appl 272:133–154, 2019). Here the adjoint operator $$T_\omega ^*$$ T ω ∗ is described. In the case where $$p=2$$ p = 2 and $$\omega $$ ω has poles only on the unit circle $${\mathbb {T}}$$ T , a description is given for when $$T_\omega ^*$$ T ω ∗ is symmetric and when $$T_\omega ^*$$ T ω ∗ admits a selfadjoint extension. If in addition $$\omega $$ ω is proper, it is shown that $$T_\omega ^*$$ T ω ∗ coincides with the unbounded Toeplitz operator defined by Sarason (Integr Equ Oper Theory 61:281–298, 2008) and studied further by Rosenfeld (Classes of densely defined multiplication and Toeplitz operators with applications to extensions of RKHS’s, 2013; J Math Anal Appl 440:911–921, 2016).

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