Mathematical Analysis
A Beurling type theorem in weighted Bergman spaces
Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 433-436.

For the vector-valued Hardy space H2(U) and the standard weighted Bergman space An(Y) with coefficient Hilbert spaces U and Y, we single out a class of contractive multipliers from H2(U) to An(Y) which we call partially isometric multipliers. We then show that a closed subspace MAn(Y) is invariant under the shift operator Sn:f(z)zf(z) if and only if M=ΦH2(U) for some partially isometric multiplier Φ from H2(U) to An(Y).

Soit H2(U) lʼespace de Hardy aux valeurs vectorielles et soit An(Y) lʼespace de Bergman aux valeurs vectorielles et au poids (1|z|2)n2, où les espaces des coefficients U et Y sont des espaces de Hilbert. Nous considérons une classe de multiplicateurs contractifs de H2(U) dans An(Y), que nous appelons multiplicateurs isométriques partiels. Nous montrons quʼun sous-espace MAn(Y) qui est invariant pour lʼoperateur Sn:f(z)zf(z) est inclus isometriquement dans An(Y) si et seulement si M=ΦH2(U) pour un multiplicateur isométrique partiel Φ de H2(U) dans An(Y).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.06.004
Ball, Joseph A. 1; Bolotnikov, Vladimir 2

1 Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA
2 Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA
@article{CRMATH_2013__351_11-12_433_0,
     author = {Ball, Joseph A. and Bolotnikov, Vladimir},
     title = {A {Beurling} type theorem in weighted {Bergman} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {433--436},
     publisher = {Elsevier},
     volume = {351},
     number = {11-12},
     year = {2013},
     doi = {10.1016/j.crma.2013.06.004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.06.004/}
}
TY  - JOUR
AU  - Ball, Joseph A.
AU  - Bolotnikov, Vladimir
TI  - A Beurling type theorem in weighted Bergman spaces
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 433
EP  - 436
VL  - 351
IS  - 11-12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.06.004/
DO  - 10.1016/j.crma.2013.06.004
LA  - en
ID  - CRMATH_2013__351_11-12_433_0
ER  - 
%0 Journal Article
%A Ball, Joseph A.
%A Bolotnikov, Vladimir
%T A Beurling type theorem in weighted Bergman spaces
%J Comptes Rendus. Mathématique
%D 2013
%P 433-436
%V 351
%N 11-12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.06.004/
%R 10.1016/j.crma.2013.06.004
%G en
%F CRMATH_2013__351_11-12_433_0
Ball, Joseph A.; Bolotnikov, Vladimir. A Beurling type theorem in weighted Bergman spaces. Comptes Rendus. Mathématique, Volume 351 (2013) no. 11-12, pp. 433-436. doi : 10.1016/j.crma.2013.06.004. http://www.numdam.org/articles/10.1016/j.crma.2013.06.004/

[1] Agler, J. Hypercontractions and subnormality, J. Oper. Theory, Volume 13 (1985) no. 2, pp. 203-217

[2] Aleman, A.; Richter, S.; Sundberg, C. Beurlingʼs theorem for the Bergman space, Acta Math., Volume 177 (1996), pp. 275-310

[3] Ball, J.A.; Bolotnikov, V. Weighted Bergman spaces: shift-invariant subspaces and input/state/output linear systems, Integral Equ. Oper. Theory, Volume 76 (2013), pp. 301-356

[4] Beurling, A. On two problems concerning linear transformations in Hilbert space, Acta Math., Volume 81 (1949), pp. 239-255

[5] Halmos, P.R. Shifts on Hilbert spaces, J. Reine Angew. Math., Volume 208 (1961), pp. 102-112

[6] Hedenmalm, H.; Perdomo, Y.G. Mean value surfaces with prescribed curvature form, J. Math. Pures Appl., Volume 83 (2004), pp. 1075-1107

[7] McCullough, S.; Trent, T.T. Invariant subspaces and Nevanlinna–Pick kernels, J. Funct. Anal., Volume 178 (2000), pp. 226-249

[8] Olofsson, A. A characteristic operator function for the class of n-hypercontractions, J. Funct. Anal., Volume 236 (2006), pp. 517-545

[9] Shimorin, S. On Beurling-type theorems in weighted 2 and Bergman spaces, Proc. Amer. Math. Soc., Volume 131 (2003) no. 6, pp. 1777-1787

Cited by Sources: