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Search: MSC category 47B35 ( Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] )

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1. CMB Online first

Marković, Marijan
A Sharp Constant for the Bergman Projection
For the Bergman projection operator $P$ we prove that \begin{equation*} \|P\colon L^1(B,d\lambda)\rightarrow B_1\| = \frac {(2n+1)!}{n!}. \end{equation*} Here $\lambda$ stands for the hyperbolic metric in the unit ball $B$ of $\mathbb{C}^n$, and $B_1$ denotes the Besov space with an adequate semi--norm. We also consider a generalization of this result. This generalizes some recent results due to Perälä.

Keywords:Bergman projections, Besov spaces
Categories:45P05, 47B35

2. CMB 2013 (vol 57 pp. 270)

Didas, Michael; Eschmeier, Jörg
Derivations on Toeplitz Algebras
Let $H^2(\Omega)$ be the Hardy space on a strictly pseudoconvex domain $\Omega \subset \mathbb{C}^n$, and let $A \subset L^\infty(\partial \Omega)$ denote the subalgebra of all $L^\infty$-functions $f$ with compact Hankel operator $H_f$. Given any closed subalgebra $B \subset A$ containing $C(\partial \Omega)$, we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra $\mathcal(B) \subset B(H^2(\Omega))$. In particular, we show that every derivation on $\mathcal{T}(A)$ is inner. These results are new even for $n=1$, where it follows that every derivation on $\mathcal{T}(H^\infty+C)$ is inner, while there are non-inner derivations on $\mathcal{T}(H^\infty+C(\partial \mathbb{B}_n))$ over the unit ball $\mathbb{B}_n$ in dimension $n\gt 1$.

Keywords:derivations, Toeplitz algebras, strictly pseudoconvex domains
Categories:47B47, 47B35, 47L80

3. CMB 2011 (vol 55 pp. 441)

Zorboska, Nina
Univalently Induced, Closed Range, Composition Operators on the Bloch-type Spaces
While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces $B^{\alpha}$ with $\alpha \ne 1$ are the ones induced by a disc automorphism.

Keywords:composition operators, Bloch-type spaces, closed range, univalent
Categories:47B35, 32A18

4. CMB 2011 (vol 54 pp. 255)

Dehaye, Paul-Olivier
On an Identity due to Bump and Diaconis, and Tracy and Widom
A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener-Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump-Diaconis-Tracy-Widom identity is a differentiated version of the classical Jacobi-Trudi identity.

Keywords:Toeplitz matrices, Jacobi-Trudi identity, Szegő limit theorem, Heine identity, Wiener-Hopf factorization
Categories:47B35, 05E05, 20G05

5. CMB 2005 (vol 48 pp. 607)

Park, Efton
Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras
For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle.

Keywords:Toeplitz operators, irrational rotation algebras, index theory
Categories:47B35, 46L80

6. CMB 2005 (vol 48 pp. 251)

Murphy, G. J.
The Index Theory Associated to a Non-Finite Trace on a $C^\ast$-Algebra
The index theory considered in this paper, a generalisation of the classical Fredholm index theory, is obtained in terms of a non-finite trace on a unital $C^\ast$-algebra. We relate it to the index theory of M.~Breuer, which is developed in a von~Neumann algebra setting, by means of a representation theorem. We show how our new index theory can be used to obtain an index theorem for Toeplitz operators on the compact group $\mathbf{U}(2)$, where the classical index theory does not give any interesting result.

Categories:46L, 47B35, 47L80

7. CMB 2002 (vol 45 pp. 309)

Xia, Jingbo
Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant
A well-known theorem of Sarason [11] asserts that if $[T_f,T_h]$ is compact for every $h \in H^\infty$, then $f \in H^\infty + C(T)$. Using local analysis in the full Toeplitz algebra $\calT = \calT (L^\infty)$, we show that the membership $f \in H^\infty + C(T)$ can be inferred from the compactness of a much smaller collection of commutators $[T_f,T_h]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper $C^\ast$-subalgebra $\calT (\calL)$ of $\calT$ which has the same essential commutant as that of $\calT$. Thus the image of $\calT (\calL)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no {\it separable} subalgebra $\calS$ of $\calT$ is capable of conferring the membership $f \in H^\infty + C(T)$ through the compactness of the commutators $\{[T_f,S] : S \in \calS\}$.

Categories:46H10, 47B35, 47C05

8. CMB 1998 (vol 41 pp. 196)

Nakazi, Takahiko
Brown-Halmos type theorems of weighted Toeplitz operators
The spectra of the Toeplitz operators on the weighted Hardy space $H^2(Wd\th/2\pi)$ and the Hardy space $H^p(d\th/2\pi)$, and the singular integral operators on the Lebesgue space $L^2(d\th/2\pi)$ are studied. For example, the theorems of Brown-Halmos type and Hartman-Wintner type are studied.

Keywords:Toeplitz operator, singular integral, operator, weighted Hardy space, spectrum.
Category:47B35

9. CMB 1998 (vol 41 pp. 240)

Xia, Jingbo
On certain $K$-groups associated with minimal flows
It is known that the Toeplitz algebra associated with any flow which is both minimal and uniquely ergodic always has a trivial $K_1$-group. We show in this note that if the unique ergodicity is dropped, then such $K_1$-group can be non-trivial. Therefore, in the general setting of minimal flows, even the $K$-theoretical index is not sufficient for the classification of Toeplitz operators which are invertible modulo the commutator ideal.

Categories:46L80, 47B35, 47C15

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