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Quasi-similarity of contractions having a 2 × 1 characteristic function. / Bermudo, Sergio; Mancera, Carmen H.; Paúl, Pedro J.; Vasyunin, Vasily.
в: Revista Matematica Iberoamericana, Том 23, № 2, 01.01.2007, стр. 677-704.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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TY - JOUR
T1 - Quasi-similarity of contractions having a 2 × 1 characteristic function
AU - Bermudo, Sergio
AU - Mancera, Carmen H.
AU - Paúl, Pedro J.
AU - Vasyunin, Vasily
PY - 2007/1/1
Y1 - 2007/1/1
N2 - Let T1 ∈ B(H1) be a completely non-unitary contraction having a non-zero characteristic function Θ1 which is a 2 × 1 column vector of functions in H∞. As it is well-known, such a function Θ1 can be written as Θ1 = w1m1[a1 b1] where w1,m1,a1,b1 ∈ H ∞ are such that W1 is an outer function with |w 1| ≤ 1, m1 is an inner function, |a1| 2 + |b1|2 = 1, and a1 = 1 (here ∧ stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction T2 ∈ B(H2) having also a 2 × 1 characteristic function Θ2 = w 2m2[a2 b2]. We prove that T 2 is quasi-similar to T2 if, and only if, the following conditions hold: 1. m1 = m2, 2. {z ∈ : |w 1(z)| < 1} = {z ∈ : |w2(z)| < 1} a.e., and 3. the ideal generated by a1 and b1 in the Smirnov class N+ equals the corresponding ideal generated by a2 and b2.
AB - Let T1 ∈ B(H1) be a completely non-unitary contraction having a non-zero characteristic function Θ1 which is a 2 × 1 column vector of functions in H∞. As it is well-known, such a function Θ1 can be written as Θ1 = w1m1[a1 b1] where w1,m1,a1,b1 ∈ H ∞ are such that W1 is an outer function with |w 1| ≤ 1, m1 is an inner function, |a1| 2 + |b1|2 = 1, and a1 = 1 (here ∧ stands for the greatest common inner divisor). Now consider a second completely non-unitary contraction T2 ∈ B(H2) having also a 2 × 1 characteristic function Θ2 = w 2m2[a2 b2]. We prove that T 2 is quasi-similar to T2 if, and only if, the following conditions hold: 1. m1 = m2, 2. {z ∈ : |w 1(z)| < 1} = {z ∈ : |w2(z)| < 1} a.e., and 3. the ideal generated by a1 and b1 in the Smirnov class N+ equals the corresponding ideal generated by a2 and b2.
KW - Characteristic functions
KW - Contractions
KW - Function models
KW - Quasi-similarity
UR - http://www.scopus.com/inward/record.url?scp=36248953203&partnerID=8YFLogxK
U2 - 10.4171/RMI/509
DO - 10.4171/RMI/509
M3 - Article
AN - SCOPUS:36248953203
VL - 23
SP - 677
EP - 704
JO - Revista Matematica Iberoamericana
JF - Revista Matematica Iberoamericana
SN - 0213-2230
IS - 2
ER -
ID: 49879834