# Multiplicative isometries on the Smirnov class

Open Mathematics (2011)

• Volume: 9, Issue: 5, page 1051-1056
• ISSN: 2391-5455

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## Abstract

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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) = $\overline{{f}^{\circ }\overline{\varphi }}$ 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) = $\left({\lambda }_{1}{z}_{{i}_{1}},...,{\lambda }_{n}{z}_{{i}_{n}}\right)$ 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.

## How to cite

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Osamu Hatori, and Yasuo Iida. "Multiplicative isometries on the Smirnov class." Open Mathematics 9.5 (2011): 1051-1056. <http://eudml.org/doc/269758>.

@article{OsamuHatori2011,
abstract = {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.},
author = {Osamu Hatori, Yasuo Iida},
journal = {Open Mathematics},
keywords = {Isometries; Smirnov class; Multiplicative maps; multiplicative maps; isometries},
language = {eng},
number = {5},
pages = {1051-1056},
title = {Multiplicative isometries on the Smirnov class},
url = {http://eudml.org/doc/269758},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Osamu Hatori
AU - Yasuo Iida
TI - Multiplicative isometries on the Smirnov class
JO - Open Mathematics
PY - 2011
VL - 9
IS - 5
SP - 1051
EP - 1056
AB - 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.
LA - eng
KW - Isometries; Smirnov class; Multiplicative maps; multiplicative maps; isometries
UR - http://eudml.org/doc/269758
ER -

## References

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2. [2] Fleming R.J., Jamison J.E., Isometries on Banach Spaces: Function Spaces, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., 129, Chapman & Hall/CRC, Boca Raton, 2003 Zbl1011.46001
3. [3] Fleming R.J., Jamison J.E., Isometries on Banach Spaces. Vol.2: Vector-Valued Function Spaces, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., 138, Chapman & Hall/CRC, Boca Raton, 2008 Zbl0367.46026
4. [4] Mazur S., Ulam S., Sur les transformations isométriques d’espaces vectoriels, normés, C. R. Acad. Sci. Paris, 1932, 194, 946–948 Zbl58.0423.01
5. [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. [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 Zbl0326.30025
7. [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 Zbl1046.46017

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