Multiplicative isometries on the Smirnov class

• Autorzy: Osamu Hatori , Yasuo Iida
• Języki publikacji: EN

Abstrakty

EN

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = $\overline {f^\circ \bar \varphi } $ for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z 1, ..., z n) = $(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } )$ for |λ j| = 1, 1 ≤ j ≤ n, and (i 1; ..., i n)is some permutation of the integers from 1through n in the case of the n-dimensional polydisk.

Słowa kluczowe

EN

Isometries; Smirnov class; Multiplicative maps

Kategorie tematyczne

• 46B04: Isometric theory of Banach spaces
• 30H15: Nevanlinna class and Smirnov class
• 32A38: Algebras of holomorphic functions

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 5
• Strony: 1051-1056

Daty

wydano

2011-10-01

online

2011-07-26

Twórcy

autor

Osamu Hatori

• Niigata University

autor

Yasuo Iida

• Iwate Medical University

Bibliografia

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• [4] Mazur S., Ulam S., Sur les transformations isométriques d’espaces vectoriels, normés, C. R. Acad. Sci. Paris, 1932, 194, 946–948
• [5] Palmer T.W., Banach Algebras and the General Theory of *-Algebras, Vol.I: Algebras and Banach Algebras, Ency-clopedia Math. Appl., 49, Cambridge University Press, Cambridge, 1994
• [6] Stephenson K., Isometries of the Nevanlinna class, Indiana Univ. Math. J., 1977, 26(2), 307–324 http://dx.doi.org/10.1512/iumj.1977.26.26023
• [7] Väisälä J., A proof of the Mazur-Ulam theorem, Amer. Math. Monthly, 2003, 110(7), 633–635 http://dx.doi.org/10.2307/3647749

Identyfikatory

DOI

10.2478/s11533-011-0068-1