Geodesic
Extended Spectra for Some Composition Operators on Weighted Hardy Spaces
Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 2, pp. 3-9

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Let $\alpha$ be a complex scalar, and let $A$ be a bounded linear operator on a Hilbert space $H$. We say that $\alpha$ is an extended eigenvalue of $A$ if there exists a nonzero bounded linear operator $X$ such that $AX=\alpha XA$. In weighted Hardy spaces invariant under automorphisms, we completely compute the extended eigenvalues of composition operators induced by linear fractional self-mappings of the unit disk $\mathbb{D}$ with one fixed point in $\mathbb{D}$ and one outside $\overline{\mathbb{D}}$. Such classes of transformations include elliptic and loxodromic mappings as well as a hyperbolic nonautomorphic mapping.
Keywords: Composition operator, extended eigenvalue, weighted Hardy space. 
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     author = {I. F. Z. Bensaid and F. Le\'on-Saavedra and P. Romero de la Rosa},
     title = {Extended {Spectra} for {Some} {Composition} {Operators} on {Weighted} {Hardy} {Spaces}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {3--9},
     publisher = {mathdoc},
     volume = {56},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2022_56_2_a0/}
}
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I. F. Z. Bensaid; F. León-Saavedra; P. Romero de la Rosa. Extended Spectra for Some Composition Operators on Weighted Hardy Spaces. Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 2, pp. 3-9. http://geodesic.mathdoc.fr/item/FAA_2022_56_2_a0/