Commutativity of \(k^{th}\)-order slant Toeplitz operators (Q2786307)

Let \(\mathbb{T}\) be the unit circle in \(\mathbb{C}\). A bounded linear operator \(U\) on \(L^2(\mathbb{T})\) is a \(k^{th}\)-order slant Toeplitz operator if and only if the relation \(M_zU=UM_{z^{k}}\) holds, where \(M_z\) and \(M_{z^k}\) denote the multiplication by \(z\) and \(z^k\), respectively. A more direct characterization of such operators \(U\) can be given in terms of its representation matrix with respect to the usual basis \(\{ z^i\mid i \in \mathbb{Z}\}\) of \(L^2(\mathbb{T})\). An operator symbol \(\varphi \in L^{\infty}(\mathbb{T})\) of \(U\) can be defined and one writes \(U=U_{\varphi}\). Given two commuting \(k^{th}\)-order slant Toeplitz operators \(U_{\varphi}\) and \(U_{\psi}\), the authors study the question how \(\varphi\) and \(\psi\) are related. It is shown that \(U_{\varphi}\) and \(U_{\psi}\) commute if and only if they commute modulo compact operators (i.e., essentially commute). Moreover, the operator product \(U_{\varphi}U_{\psi}\) is compact if and only if it is trivial. Then the (essentially) commuting operators \(U_{\varphi}\) and \(U_{\psi}\) are precisely characterized under certain additional assumptions on one of the symbols. NEWLINENEWLINEIn the second part of the paper, \(k^{th}\)-order slant Toeplitz operators \(B_{\varphi}\) acting on the Bergman space \(L_a^2(\mathbb{D})\) over the unit disc \(\mathbb{D}\subset \mathbb{C}\) and having bounded symbols \(\varphi \in L^{\infty}(\mathbb{D})\) are defined and analyzed. Similar to the previous case, \(B_{\varphi}\) can be expressed as an operator product \(W_kT_{\varphi}\), where \(W_k\) is a certain bounded operator on \(L_a^2(\mathbb{D})\) and \(T_{\varphi}\) denotes the Bergman Toeplitz operator with symbol \(\varphi\). First, it is shown that \(B_{\varphi}\) can be expressed as an integral operator and its kernel is explicitly calculated. In the case of bounded holomorphic operator symbols \(\varphi\) and \(\psi\) on \(\mathbb{D}\), the commuting pairs \((B_{\varphi}, B_{\psi})\) are characterized.