Fredholm properties of Toeplitz operators on Dirichlet spaces
The author analyzes Toeplitz operators on Dirichlet spaces and computes spectra of Toeplitz operators with symbols in \(C^1,H^\infty_1+ C^1\). Let \(D\) be the unit disk in the complex plane \(\mathbb{C}\), \(dA={1\over \pi} dx dy\) be the normalized area measure on \(D\). \(L^{2,1}\) is the space of functions \(u:D\to\mathbb{C}\), for which the norm \[ \|u\|_{{1\over 2}}= \left[\int_D \Biggl(\Biggl|{\partial u\over\partial z}(z) \Biggr|^2+ \Biggl|{\partial u\over\partial\overline z}(z)\Biggr|^2 dA\Biggr)\right]^{{1\over 2}}<\infty. \] \(L^{2,1}\) is a Hilbert space with the inner product \[ \langle u,v\rangle_{{1\over 2}}= \Biggl\langle{\partial u\over\partial z}, {\partial v\over\partial z}\Biggr\rangle_{L^2(dA)}+ \Biggl\langle{\partial u\over\partial\overline z}, {\partial v\over\partial\overline z}\Biggr\rangle_{L^2(dA)}. \] The Dirichlet space \({\mathbf D}\) is the subspace of all analytic functions \(g\) in \(L^{2,1}\) with \(g(0)= 0\). Let \(P\) be the orthogonal projection from \(L^{2,1}\) into \({\mathbf D}\). \(P\) is an integral operator defined by \[ P(u)(w)= \int_D {\partial u\over\partial z} {\overline{\partial K(z,w)}\over\partial z} dA, \] where \(K(z,w)= \sum^\infty_{k= 1}{z^kw^k\over k}\) is the reproducing kernel of \({\mathbf D}\). Definition 1. Suppose \(\varphi\in C^1(D)\), the operator \[ T_\varphi f(w)= P(\varphi f, K)_{{1\over 2}}= \int_D {\partial(\varphi f)\over\partial z} {\overline{\partial K(z,w)}\over\partial z} dA(z) \] is said to be Toeplitz operator with symbol \(\varphi\). In the first section of the article, the author computes the spectra and essential spectra of Toeplitz operators with symbol \(\varphi\). In the next section, the author proves that \(H^\infty_1+ C^1(D)\) is a norm-closed space relative to a suitable norm, and obtains the representation of essential spectra of the Toeplitz operators with symbols \(H^\infty_1+ C^1(D)\).
scientific article; zbMATH DE number 1344303