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Tohoku Mathematical Journal
SECOND SERIES VOL. 68, NO. 2
|Tohoku Math. J.|
68 (2016), 253-271
COMPACT SUMS OF TOEPLITZ PRODUCTS AND TOEPLITZ ALGEBRA ON THE DIRICHLET SPACE
Young Joo Lee
(Received August 4, 2014, revised October 21, 2014)
In this paper we consider Toeplitz operators on the Dirichlet space of the ball. We first characterize the compactness of operators which are finite sums of products of two Toeplitz operators. We also characterize Fredholm Toeplitz operators and describe the essential norm of Toeplitz operators. By using these results, we establish a short exact sequence associated with the $C^*$-algebra generated by all Toeplitz operators.
Mathematics Subject Classification.
Primary 47B35; Secondary 32A37.
Key words and phrases.
Toeplitz operator, Dirichlet space, Fredholm operators, Toeplitz algebra.
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