Standard

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.

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Harvard

Bermudo, S, Mancera, CH, Paúl, PJ & Vasyunin, V 2007, 'Quasi-similarity of contractions having a 2 × 1 characteristic function', Revista Matematica Iberoamericana, Том. 23, № 2, стр. 677-704. https://doi.org/10.4171/RMI/509

APA

Bermudo, S., Mancera, C. H., Paúl, P. J., & Vasyunin, V. (2007). Quasi-similarity of contractions having a 2 × 1 characteristic function. Revista Matematica Iberoamericana, 23(2), 677-704. https://doi.org/10.4171/RMI/509

Vancouver

Bermudo S, Mancera CH, Paúl PJ, Vasyunin V. Quasi-similarity of contractions having a 2 × 1 characteristic function. Revista Matematica Iberoamericana. 2007 Янв. 1;23(2):677-704. https://doi.org/10.4171/RMI/509

Author

Bermudo, Sergio ; Mancera, Carmen H. ; Paúl, Pedro J. ; Vasyunin, Vasily. / Quasi-similarity of contractions having a 2 × 1 characteristic function. в: Revista Matematica Iberoamericana. 2007 ; Том 23, № 2. стр. 677-704.

BibTeX

@article{e5efd8e207d14fb395374aabeeb73528,
title = "Quasi-similarity of contractions having a 2 × 1 characteristic function",
abstract = "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.",
keywords = "Characteristic functions, Contractions, Function models, Quasi-similarity",
author = "Sergio Bermudo and Mancera, {Carmen H.} and Pa{\'u}l, {Pedro J.} and Vasily Vasyunin",
year = "2007",
month = jan,
day = "1",
doi = "10.4171/RMI/509",
language = "English",
volume = "23",
pages = "677--704",
journal = "Revista Matematica Iberoamericana",
issn = "0213-2230",
publisher = "Universidad Autonoma de Madrid",
number = "2",

}

RIS

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