Operators on the orthogonal complement of the Dirichlet space
The purpose of this paper is the study of some properties of dual Hankel and Toeplitz operators on the orthogonal complement of the Dirichlet space of the open unit disk \(\mathbb{D}\). The Sobolev space \(W^{1,2}\) is a Hilbert space of functions with the inner product \[ \langle u,\nu\rangle=\int_{\mathbb{D}}u \,dA\int_{\mathbb{D}}\bar{\nu}\,dA+\left\langle\frac{\partial u}{\partial z},\frac{\partial\nu}{\partial z}\right\rangle_2+\left\langle\frac{\partial u}{\partial\bar{z}}, \frac{\partial \nu}{\partial\bar{z}}\right\rangle_2. \] Here, \(\langle \cdot,\cdot\rangle_2\) represents the inner product in \(L^2(\mathbb{D},dA)\), \(dA\) denotes the normalized Lebesgue measure on \(\mathbb{D}\), and derivatives are considered in the weak sense. The Dirichlet space \(\mathcal{D}\) is the closed subspace of \(W^{1,2}\) consisting of all analytic functions on \(\mathbb{D}\) vanishing at \(0\). Let \(P\) denote the orthogonal projection from \(W^{1,2}\) onto \(\mathcal{D}\). For a function \(\phi\in W^{1,\infty}=\{u\in W^{1,2}:u,\frac{\partial u}{\partial z}, \frac{\partial u}{\partial\bar{z}}\in L^{\infty}(\mathbb{D})\}\), the Toeplitz operator \(T_{\phi}:\mathcal{D}\rightarrow\mathcal{D}\), the Hankel operator \(H_{\phi}:\mathcal{D}\rightarrow\mathcal{D}^{\perp}\), the dual Toeplitz operator \(S_{\phi}:\mathcal{D}^{\perp}\rightarrow \mathcal{D}^{\perp}\) and the dual Hankel operator \(R_{\phi}:\mathcal{D}^{\perp}\rightarrow\mathcal{D}\) are defined, respectively, by \[ T_{\phi}h=P(\phi h), \;H_{\phi}(h)=(I-P)(\phi h), \;S_{\phi}(u)=(I-P)(\phi u) \] and \[ R_{\phi}(u)=P(\phi u), \quad h\in\mathcal{D}, \, u\in\mathcal{D}^{\perp}. \] In the first theorem, the author proves that for \(\phi\in W^{1,\infty}\), \(R_{\phi}=0\) if and only if \(\phi\in\mathcal{D}^{\perp}\). The second theorem deals with commutativity of dual Toeplitz operators. In the Bergman space setting, \textit{K.\,Stroethoff} and \textit{D.--C.\thinspace Zheng} [``Algebraic and spectral properties of dual Toeplitz operators'', Trans.\ Am.\ Math.\ Soc.\ 354, No.\,6, 2495--2520 (2002; Zbl 0996.47036)] characterized commutavity of dual Toeplitz operators. In the present paper, the author shows that the result is somewhat different: \(S_{\phi}S_{\psi}=S_{\psi}S_{\phi}\) if and only if one of the following statements holds: (1) Both \(\phi\) and \(\psi\) are analytic on \(\mathbb{D}\). (2) Both \(\phi\) and \(\psi\) are in \(\mathcal{D}^{\perp}\). (3) A nontrivial linear combination of \(\phi\) and \(\psi\) is constant on \(\mathbb{D}\). The author finishes the paper with an example of two nonzero symbols \(\phi\) and \(\psi\) in \(\mathcal{D}^{\perp}\) for which \(S_{\phi}S_{\psi}=0\). This is in sharp contrast with the zero-divisor problem for Toeplitz and dual Toeplitz operators on the Hardy and Bergman spaces.
scientific article; zbMATH DE number 5568993