ISSN:
1420-8989
Keywords:
47B35
;
47D25
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let $$\mathbb{D}$$ be the unit disk inℂ, $$\mathcal{A}^2 (\mathbb{D})$$ be the Bergman space, consisting of all analytic functions from $$L_2 (\mathbb{D})$$ , and $$B_\mathbb{D} $$ be the Bergman projection of $$L_2 (\mathbb{D})$$ onto $$\mathcal{A}^2 (\mathbb{D})$$ . We constructC *-algebras $$\mathcal{A} \subset L_\infty (\mathbb{D})$$ , for functions of which the commutator of Toeplitz operators [T a ,T b ]=T a T b −T b T a is compact, and, at the same time, the semi-commutator [T a ,T b )=T a T b −T ab is not compact. It is proved, that for each finite set ∧=〈n 0,n 1, ...,n m 〉, where 1=n 0 〈n 1 〈...〈n m ≤∞, andn k ∈ℕ∪ {∞}, there are algebras $$\mathcal{A}_\Lambda $$ of the above type, such that the symbol algebras Sym $$\mathcal{T}(\mathcal{A}_\Lambda )$$ of Toeplitz operator algebras $$\mathcal{T}(\mathcal{A}_\Lambda )$$ arecommutative, while the symbol algebras Sym $$\mathcal{R}(\mathcal{A}_\Lambda ,B_\mathbb{D} )$$ of the algebras $$\mathcal{R}(\mathcal{A}_\Lambda ,B_\mathbb{D} )$$ , generated by multiplication operators $$a \in \mathcal{A}_\Lambda $$ and $$B_\mathbb{D} $$ , haveirreducible representations exactly of dimensions n 0,n 1,..., n m .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01332495
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