Commutative algebras of Toeplitz operators on the Bergman space revisited: spectral theorem approach (Q2163590)

Let \(\mathbb{D}\) be the open unit disk, \(\mathbb{T}=\partial\mathbb{D}\), and let \(\Pi\) be the upper half-plane. The paper deals with studying a bijection between commutative \(C^*\)-algebras generated by Toeplitz operators with bounded measurable symbols that act on the standard weighted Bergman space \({\mathcal A}^2_\lambda(\mathbb{D})\) \((\lambda>-1)\) and maximal abelian subgroups of Möbius transforms of \(\mathbb{D}\). Each one-parameter abelian subgroup \(G\) of this type is conjugated to one of the following three model groups: \begin{itemize} \item elliptic, \(\mathbb{T}:\mathbb{D}\to\mathbb{D}\), with action \(t\in\mathbb{T}:z\mapsto tz\); \item parabolic, \(\mathbb{R}:\Pi\to\Pi\), with action \(h\in\mathbb{R}:z\mapsto z+h\); \item hyperbolic, \(\mathbb{R}_+:\Pi\to\Pi\), with action \(\rho\in\mathbb{R}_+:z\mapsto\rho z\). \end{itemize} The orbits of these groups are given by the following parametric equations: \begin{itemize} \item group \(\mathbb{T}\) on \(\mathbb{D}\), \(x=R\cos\theta\), \(y=R\sin\theta\), \(\theta\in[0,2\pi)\), with a fixed \(R\in(0,1)\); \item group \(\mathbb{R}\) on \(\Pi\): \(x=h\), \(y=y_0\), \(h\in\mathbb{R}\), with a fixed \(y_0\in\mathbb{R}_+\); \item group \(\mathbb{R}_+\) on \(\Pi\): \(x=\rho\cos\theta\), \(y=\rho\sin\theta\), \(\rho\in\mathbb{R}_+\), with a fixed \(\theta\in(0,\pi)\). \end{itemize} These orbits are integral curves, respectively, for the following vector fields: \[ V_{\mathrm{ell}}=-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\quad V_{\mathrm{par}}=\frac{\partial}{\partial x},\quad V_{\mathrm{hyp}}= x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}. \] For each \(G\in\{\mathbb{T},\mathbb{R},\mathbb{R}_+\}\), there exists a Borel subset \(X_G\subset\mathbb{R}\) with a measure \(\sigma_G\) and a unitary operator \(R_G:{\mathcal A}^2_\lambda\to L_2(X_G,\sigma_G)\) such that each Toeplitz operator \(T_a\) with \(G\)-invariant symbol \(a\) is unitarily equivalent to the multiplication operator by a certain function \(\gamma_a\) on the space \(L_2(X_G,\sigma_G)\), \[ R_GT_aR_G^*=\gamma_aI. \] The map \(T_a\mapsto\gamma_a\) defined initially for Toeplitz operators with bounded symbols \(a\) can be extended to bounded Toeplitz operators with wider classes of symbols \(a\) by using the following approach. With each model commutative algebra, the authors associate a certain self-adjoint (unbounded) operator \(N\) which generates this algebra via the functional calculus based on the Spectral Theorem. For example, for \(G=\mathbb{R}\), they construct the spectral measure \(E\) of \(N\) such that the operator \(\varphi(N)=\int_{\mathbb{R}} \varphi(\eta)dE(\eta)\) is well defined and normal on a domain \({\mathcal D}_\varphi \subset{\mathcal A}^2_\lambda(\Pi)\) for every \(E\)-measurable function \(\varphi\). The operator \(\varphi(N)\) is bounded, and thus defined on the whole \({\mathcal A}^2_\lambda(\Pi)\), if and only if the function \(\varphi\) is \(E\)-essentially bounded. The mapping \({\mathcal J}:\varphi\mapsto\varphi(N)\) is an isometric isomorphism of the unital \(C^*\)-algebra \(L_\infty(\mathbb{R},E)\) with involution \(\varphi\mapsto \overline{\varphi}\) onto a commutative unital algebra of bounded linear operators on the space \({\mathcal A}^2_\lambda(\Pi)\) with involution \(H\mapsto H^*\), and \({\mathcal J} (L_\infty(\mathbb{R},E))\) is a von Neumann algebra. Relations between properties of the spectral functions \(\gamma_a\) of Toeplitz operators \(T_a\), their symbols \(a\) and the spectral representations of the considered commutative algebras of Toeplitz operators are studied.