Toeplitz algebra and Hankel algebra on the harmonic Bergman space (Q1856976)

The authors study the Toeplitz algebra \(\mathcal{T}\) and the Hankel algebra \(\mathcal{H}\) on the harmonic Bergman space \(L^2_h\) and give a characterization of compact Toeplitz operators. Let \(\mathcal{K}\) be the ideal of compact operators on \(L^2_h\) and \(C(T)\) the algebra of continuous functions on the unit circle \(T\). It is shown that the sequence \(0 \rightarrow {\mathcal{K}}\rightarrow{\mathcal{T}}@>\pi>> C(T)\rightarrow 0\) is exact where \(\pi (T_f+{\mathcal{K}})=f |_T\) on \(\mathcal{T}/\mathcal{K},\) and \(\operatorname {ind} A=0\) for each Fredholm operator \(A\) in \(\mathcal{H}\).