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Hermitian weighted composition operators and Bergman extremal functions. (English) Zbl 1296.47025
For an analytic function $$f$$ on the unit disk $$\mathbb D$$ and an analytic self-map $$\varphi$$ of $$\mathbb D$$, let $$W_{f, \varphi}$$ be the weighted composition operator defined by $$W_{f, \varphi}h = f (h\circ \varphi)$$ for functions $$h$$ analytic on $$\mathbb D$$. Let $$H^2(\beta)$$ be the weighted Hardy space over $$\mathbb D$$ with weight sequence $$\beta$$. The weighted Hardy space whose reproducing kernel is $$(1-z\bar w )^{-\kappa}$$, $$\kappa\geq 1$$, is denoted by $$H^2(\beta_\kappa)$$. So, $$H^2(\beta_\kappa)$$ is the Hardy space for $$\kappa=1$$ and is the standard weighted Bergman space for $$\kappa>1$$.
The authors first observe necessary conditions for $$W_{f, \varphi}$$ to be Hermitian and bounded on a given $$H^2(\beta)$$. The necessary conditions assert that $$f$$ and $$\varphi$$ must be linear fractional maps of special type. Those necessary conditions are also shown to be sufficient when the spaces are restricted to $$H^2(\beta_\kappa)$$, $$\kappa \geq -1$$. Moreover, as in the Hardy space case due to two of the present authors [C. C. Cowen and E. Ko, Trans. Am. Math. Soc. 362, No. 11, 5771–5801 (2010; Zbl 1213.47034)], they show on $$H^2(\beta_\kappa)$$ that the Hermitian weighted composition operators are divided into three classes: (i) compact operators, (ii) multiples of isometries, and (iii) those that have no eigenvalues. In the first two cases, the eigenvalues and eigenvectors are explicitly computed, and in the third case additional properties of being cyclic and being part of an analytic semigroup that includes normal weighted composition operators are proved. Finally, for the weighted Bergman spaces $$H^2(\beta_\kappa)$$, $$\kappa>1$$, the authors apply one of their results in the present paper to compute the extremal functions for the invariant subspaces associated with the usual atomic inner functions $$e^{c {{1+z}\over{1-z}}}$$, $$c<0$$, and obtain explicit formulas for the projections of the reproducing kernel functions to those invariant subspaces. Those extremal functions and projected reproducing kernel functions are explicitly described in terms of the incomplete Gamma function.

##### MSC:
 47B33 Linear composition operators 47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B38 Linear operators on function spaces (general) 30H20 Bergman spaces and Fock spaces
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