Multiplicative isometries on the Smirnov class
Open Mathematics (2011)
- Volume: 9, Issue: 5, page 1051-1056
- ISSN: 2391-5455
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topOsamu 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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