scholarly journals COMPACT TOEPLITZ OPERATORS WITH CONTINUOUS SYMBOLS

2009 ◽  
Vol 51 (2) ◽  
pp. 257-261 ◽  
Author(s):  
TRIEU LE
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Unit Sphere ◽  
Orthogonal Projection ◽  
Toeplitz Operators ◽  
Borel Measure ◽  
Borel Function ◽  
Closed Unit Ball ◽  
Rotation Invariant ◽  
Regular Borel Measure

AbstractFor any rotation-invariant positive regular Borel measure ν on the closed unit ball $\overline{\mathbb{B}}_n$ whose support contains the unit sphere $\mathbb{S}_n$, let L2a be the closure in L2 = L2($\overline{\mathbb{B}}_n, dν) of all analytic polynomials. For a bounded Borel function f on $\overline{\mathbb{B}}_n$, the Toeplitz operator Tf is defined by Tf(ϕ) = P(fϕ) for ϕ ∈ L2a, where P is the orthogonal projection from L2 onto L2a. We show that if f is continuous on $\overline{\mathbb{B}}_n$, then Tf is compact if and only if f(z) = 0 for all z on the unit sphere. This is well known when L2a is replaced by the classical Bergman or Hardy space.

scholarly journals Remarks on retracting balls on spherical caps in \(c_{0}\), \(c\), \(l^{\infty }\) spaces

2020 ◽  
Vol 74 (1) ◽  
pp. 45
Author(s):  
Kazimierz Goebel
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Unit Sphere ◽  
Uniform Norm ◽  
Sequence Spaces ◽  
Closed Unit Ball ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Lipschitz Constants ◽  
Spherical Caps

For any infinite dimensional Banach space there exists a lipschitzian retraction of the closed unit ball B onto the unit sphere S. Lipschitz constants for such retractions are, in general, only roughly estimated. The paper is illustrative. It contains remarks, illustrations and estimates concerning optimal retractions onto spherical caps for sequence spaces with the uniform norm.

scholarly journals Functions invariant under the Bochner–Martinelli integral

2002 ◽  
Vol 65 (1) ◽  
pp. 55-57
Author(s):  
Jaesung Lee
Keyword(s):  
Unit Ball ◽  
Unit Sphere ◽  
Elementary Proof ◽  
Integral Transform ◽  
Closed Unit Ball

We give an elementary proof of the statement that a function f on the closed unit ball of Cn, integrable on the unit sphere, is holomorphic if it is invariant under the Bochner–Martinelli integral transform.

scholarly journals Midpoint sets contained in the unit sphere of a normed space

2011 ◽  
Vol 48 (2) ◽  
pp. 180-192
Author(s):  
Konrad Swanepoel
Keyword(s):  
Unit Ball ◽  
Unit Sphere ◽  
Normed Space ◽  
Higher Dimensions ◽  
Three Dimensions ◽  
Closed Unit Ball ◽  
Facial Structure ◽  
Finite Dimensional ◽  
Dimensional Normed Space

The midpoint set M(S) of a set S of points is the set of all midpoints of pairs of points in S. We study the largest cardinality of a midpoint set M(S) in a finite-dimensional normed space, such that M(S) is contained in the unit sphere, and S is outside the closed unit ball. We show in three dimensions that this maximum (if it exists) is determined by the facial structure of the unit ball. In higher dimensions no such relationship exists. We also determine the maximum for euclidean and sup norm spaces.

Singular Integral Operators and Essential Commutativity on the Sphere

2010 ◽  
Vol 62 (4) ◽  
pp. 889-913 ◽  
Author(s):  
Jingbo Xia
Keyword(s):  
Hardy Space ◽  
Singular Integral ◽  
Unit Sphere ◽  
Toeplitz Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Essential Commutant

AbstractLet 𝒯 be the C*-algebra generated by the Toeplitz operators {T𝜑 : 𝜑 Є L∞(S, dσ)} on the Hardy space H2(S) of the unit sphere in Cn. It is well known that 𝒯 is contained in the essential commutant of {T𝜑 : 𝜑 Є VMO∩L∞(S, dσ)}. We show that the essential commutant of {T𝜑 : 𝜑 Є VMO∩L∞(S, dσ)} is strictly larger than 𝒯.

scholarly journals On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

2021 ◽  
Author(s):  
Mahamet Koïta ◽  
Stanislas Kupin ◽  
Sergey Naboko ◽  
Belco Touré
Keyword(s):  
Spectral Analysis ◽  
Continuous Function ◽  
Bergman Space ◽  
Orthogonal Projection ◽  
Spectral Properties ◽  
Toeplitz Operators ◽  
Closed Subspace ◽  
Jacobi Matrices ◽  
Banded Matrices ◽  
Eigen Values

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

Weak-Star Continuous Analytic Functions

1995 ◽  
Vol 47 (4) ◽  
pp. 673-683 ◽  
Author(s):  
R. M. Aron ◽  
B. J. Cole ◽  
T. W. Gamelin
Keyword(s):  
Unit Ball ◽  
Finite Type ◽  
Analytic Functions ◽  
Approximation Property ◽  
Entire Functions ◽  
Open Unit ◽  
Closed Unit Ball ◽  
Open Unit Ball ◽  
Weakly Continuous ◽  
Weak Star

AbstractLet 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

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