A note concerning the index of the shift
Author:
John R. Akeroyd
Journal:
Proc. Amer. Math. Soc. 130 (2002), 33493354
MSC (2000):
Primary 47A53, 47B20, 47B38; Secondary 30E10, 46E15
Published electronically:
April 11, 2002
MathSciNet review:
1913014
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let be a finite, positive Borel measure with support in such that  the closure of the polynomials in  is irreducible and each point in is a bounded point evaluation for . We show that if and there is a nontrivial subarc of such that
then for each nontrivial closed invariant subspace for the shift on .
 [A1]
J. Akeroyd, Another look at some index theorems for the shift, Indiana Univ. Math. J., vol. 50, no. 2 (2001), 705718.
 [A2]
J. Akeroyd, An Extension of Szegö's Theorem II, Indiana Univ. Math. J., vol. 45, no. 1 (1996), 241252. MR 97h:30055
 [ABFP]
C. Apostol, H. Bercovici, C. Foias, C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I, J. of Funct. Analysis 63 (1985), 369404. MR 87i:47004a
 [C]
J. B. Conway, The Theory of Subnormal Operators, Math. Surveys Monographs, Vol. 36 (1991), Amer. Math. Soc., Providence, RI. MR 92h:47026
 [CY]
J. B. Conway, L. Yang, Some open problems in the theory of subnormal operators, Holomorphic spaces, Cambridge University Press, vol. 33, 1998, 201209. MR 99e:47027
 [H]
K. Hoffman, Banach Spaces of Analytic Functions, PrenticeHall, Englewood Cliffs, N.J., 1962. MR 24:A2844
 [HRS]
H. Hedenmalm, S. Richter, K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math., 477 (1996), 1330. MR 97i:46044
 [KM]
T. L. Kriete, B. D. MacCluer, Meansquare approximation by polynomials on the unit disk, Trans. Amer. Math. Soc., vol. 322, no. 1 (1990), 134. MR 91b:30119
 [M]
T. L. Miller, Some subnormal operators not in , J. Functional Analysis, 82 (1989), 296302. MR 90c:47040
 [OT]
R. F. Olin, J. E. Thomson, Some index theorems for subnormal operators, J. Operator Theory, 3 (1980), 115142. MR 81f:47031
 [OY1]
R. F. Olin, L. Yang, A subnormal operator and its dual, Canad. J. Math., vol. 48, no. 2 (1996), 381396. MR 98j:47055
 [OY2]
R. F. Olin, L. Yang, The commutant of multiplication by on the closure of the polynomials in , J. of Funct. Analysis, vol. 134, no. 2 (1995), 297320. MR 97m:47023
 [T]
J. E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), 477507. MR 93g:47026
 [TY]
J. E. Thomson, L. Yang, Invariant subspaces with the codimension one property in , Indiana Univ. Math. J., vol. 44, no. 4 (1995), 11631173. MR 97c:47036
 [Y]
L. Yang, Invariant subspaces of the Bergman space and some subnormal operators in , Mich. Math. J. 42 (1995), 301310. MR 96f:47013
 [A1]
 J. Akeroyd, Another look at some index theorems for the shift, Indiana Univ. Math. J., vol. 50, no. 2 (2001), 705718.
 [A2]
 J. Akeroyd, An Extension of Szegö's Theorem II, Indiana Univ. Math. J., vol. 45, no. 1 (1996), 241252. MR 97h:30055
 [ABFP]
 C. Apostol, H. Bercovici, C. Foias, C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I, J. of Funct. Analysis 63 (1985), 369404. MR 87i:47004a
 [C]
 J. B. Conway, The Theory of Subnormal Operators, Math. Surveys Monographs, Vol. 36 (1991), Amer. Math. Soc., Providence, RI. MR 92h:47026
 [CY]
 J. B. Conway, L. Yang, Some open problems in the theory of subnormal operators, Holomorphic spaces, Cambridge University Press, vol. 33, 1998, 201209. MR 99e:47027
 [H]
 K. Hoffman, Banach Spaces of Analytic Functions, PrenticeHall, Englewood Cliffs, N.J., 1962. MR 24:A2844
 [HRS]
 H. Hedenmalm, S. Richter, K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math., 477 (1996), 1330. MR 97i:46044
 [KM]
 T. L. Kriete, B. D. MacCluer, Meansquare approximation by polynomials on the unit disk, Trans. Amer. Math. Soc., vol. 322, no. 1 (1990), 134. MR 91b:30119
 [M]
 T. L. Miller, Some subnormal operators not in , J. Functional Analysis, 82 (1989), 296302. MR 90c:47040
 [OT]
 R. F. Olin, J. E. Thomson, Some index theorems for subnormal operators, J. Operator Theory, 3 (1980), 115142. MR 81f:47031
 [OY1]
 R. F. Olin, L. Yang, A subnormal operator and its dual, Canad. J. Math., vol. 48, no. 2 (1996), 381396. MR 98j:47055
 [OY2]
 R. F. Olin, L. Yang, The commutant of multiplication by on the closure of the polynomials in , J. of Funct. Analysis, vol. 134, no. 2 (1995), 297320. MR 97m:47023
 [T]
 J. E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), 477507. MR 93g:47026
 [TY]
 J. E. Thomson, L. Yang, Invariant subspaces with the codimension one property in , Indiana Univ. Math. J., vol. 44, no. 4 (1995), 11631173. MR 97c:47036
 [Y]
 L. Yang, Invariant subspaces of the Bergman space and some subnormal operators in , Mich. Math. J. 42 (1995), 301310. MR 96f:47013
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Additional Information
John R. Akeroyd
Affiliation:
Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
Email:
jakeroyd@comp.uark.edu
DOI:
http://dx.doi.org/10.1090/S000299390206464X
PII:
S 00029939(02)06464X
Received by editor(s):
April 17, 2001
Received by editor(s) in revised form:
June 19, 2001
Published electronically:
April 11, 2002
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2002 American Mathematical Society
