Acta mathematica scientia, Series B >
WANDERING SUBSPACES OF THE HARDY-SOBOLEV SPACES OVER Dn
Received date: 2015-01-22
Revised date: 2016-03-23
Online published: 2016-10-25
Supported by
This work was partially supported by the Natural Science Foundation of China (11271092, 11471143), the key research project of Nanhu College of Jiaxing University (N41472001-18).
In this paper, we show that for (log(2)/(3))/(2 log 2)≤β≤(1)/(2), suppose S is an invariant subspace of the Hardy-Sobolev spaces Hβ2(Dn) for the n-tuple of multiplication operators (Mz1,…, Mzn). If (Mz1|S,…, Mzn|S) is doubly commuting, then for any non-empty subset α={α1,…, αk} of {1,…,n}, WαS is a generating wandering subspace for Wα|S=(Mzα1|S,…, Mzαk|S), that is,[WαS]Wα|S=S, where WαS
(S?zαiS).
Jiesheng XIAO , Guangfu CAO . WANDERING SUBSPACES OF THE HARDY-SOBOLEV SPACES OVER Dn[J]. Acta mathematica scientia, Series B, 2016 , 36(5) : 1467 -1473 . DOI: 10.1016/S0252-9602(16)30082-0
[1] Aleman A, Richter S, Sundberg C. Beurling's theorem for the Bergman space. Acta Math, 1996, 177:275-310
[2] Beurling A. On two problems concerning linear transformations in Hilbert space. Acta Math, 1949, 81:239-255
[3] Chattopadhyay A, Das B, Sarkar J, Sarkar S. Wandering subspaces of the Bergman space and the Dirichlet space over Dn. Integr Equ Oper Theory, 2014, 79:567-577
[4] Halmos P R. Shifts on Hilbert spaces. J Reine Angew Math, 1961, 208:102-112
[5] Hedenmalm H. An invariant subspace of the Bergman space having the codimension two property. J Reine Angew Math, 1993, 443:1-9
[6] Mandrekar V. The validity of Beurling theorems in polydiscs. Proc Amer Math Soc, 1988, 103:145-148
[7] Redett D, Tung J. Invariant subspaces in Bergman space over the bidisc. Proc Amer Math Soc, 2010, 138:2425-2430
[8] Richter S. Invariant subspaces of the Dirichlet shift. J Reine Angew Math, 1988, 386:205-220
[9] Rudin W. Function Theory in Polydiscs. New York:Benjamin, 1969
[10] Shimorin S. Wold-type decompositions and wandering subspaces for operators close to isometries. J Reine Angew Math, 2001, 531:147-189
[11] Sarkar J, Sasane A, Wick B. Doubly commuting submodules of the Hardy module over polydiscs. Studia Math, 2013, 217:179-192
[12] Wang X F, Cao G F. Galerkin-Petrov methods of Toeplitz operators on Dirichlet space. Acta Math Sci, 2007, 27B(2):308-317
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