On the selfcommutator of automorphic composition operators

M. T. Heydari ()
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M. T. Heydari: Yasouj University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 1, 274-278

Abstract: Abstract Our principal interest here is in the selfcommutator and the anti selfcommutator of an automorphic composition operator $$C_{\varphi }$$ C φ which denoted by $$[C^*_{\varphi }; C_{\varphi }]$$ [ C φ ∗ ; C φ ] and $$\{C^*_{\varphi }; C_{\varphi }\}$$ { C φ ∗ ; C φ } , and defined by $$[C^*_{\varphi }; C_{\varphi }] = C^*_{\varphi }C_{\varphi }-C_{\varphi }C^*_{\varphi }$$ [ C φ ∗ ; C φ ] = C φ ∗ C φ - C φ C φ ∗ and $$\{C^*_{\varphi }; C_{\varphi }\} = C^*_{\varphi }C_{\varphi }+C_{\varphi }C^*_{\varphi }$$ { C φ ∗ ; C φ } = C φ ∗ C φ + C φ C φ ∗ respectively. In this work, we compute an explicit expression for the adjoint of a composition operator on $$H^2$$ H 2 induced by a conformal automorphism and the starting point for our analysis of $$[C^*_{\varphi }; C_{\varphi }]$$ [ C φ ∗ ; C φ ] and $$\{C^*_{\varphi }; C_{\varphi }\}$$ { C φ ∗ ; C φ } is this formula. Also, we compute the spectrum and norm of $$[C^*_{\varphi }; C_{\varphi }]$$ [ C φ ∗ ; C φ ] and $$\{C^*_{\varphi }; C_{\varphi }\}$$ { C φ ∗ ; C φ } for involution automorphism $$\varphi $$ φ .

Keywords: Selfcommutator; Composition operator; Spectrum; Adjoint; Primary 47B33; Secondary 47B47 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00251-8

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