Geodesic
Isometries and Hermitian operators on complex symmetric sequence spaces
Matematičeskie trudy, Tome 20 (2017) no. 1, pp. 21-42

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We consider a complex symmetric sequence space $E$ that possesses the Fatou property and is different from $l_2$. We prove that, for every surjective linear isometry $V$ on $E$, there exist $\lambda_n\in\mathbb C$ with $|\lambda_n|=1$ and a bijective mapping $\pi$ on the set $\mathbb N$ of natural numbers such that $$ V\big(\{\xi_n\}_{n\in\mathbb N} \big)=\big\{\lambda_n\xi_{\pi(n)} \big\}_{n\in\mathbb N} $$ for every $\{\xi_n\}_{n\in\mathbb N}\in E$. 
@article{MT_2017_20_1_a1,
     author = {B. R. Aminov and V. I. Chilin},
     title = {Isometries and {Hermitian} operators on complex symmetric sequence spaces},
     journal = {Matemati\v{c}eskie trudy},
     pages = {21--42},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2017_20_1_a1/}
}
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B. R. Aminov; V. I. Chilin. Isometries and Hermitian operators on complex symmetric sequence spaces. Matematičeskie trudy, Tome 20 (2017) no. 1, pp. 21-42. http://geodesic.mathdoc.fr/item/MT_2017_20_1_a1/