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 
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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]
John
Akeroyd, An extension of Szegő’s theorem. II,
Indiana Univ. Math. J. 45 (1996), no. 1,
241–252. MR 1406692
(97h:30055), http://dx.doi.org/10.1512/iumj.1996.45.1195
 [ABFP]
C.
Apostol, H.
Bercovici, C.
Foias, and C.
Pearcy, Invariant subspaces, dilation theory, and the structure of
the predual of a dual algebra. I, J. Funct. Anal. 63
(1985), no. 3, 369–404. MR 808268
(87i:47004a), http://dx.doi.org/10.1016/00221236(85)90093X
 [C]
John
B. Conway, The theory of subnormal operators, Mathematical
Surveys and Monographs, vol. 36, American Mathematical Society,
Providence, RI, 1991. MR 1112128
(92h:47026)
 [CY]
John
B. Conway and Liming
Yang, Some open problems in the theory of subnormal operators,
Holomorphic spaces (Berkeley, CA, 1995) Math. Sci. Res. Inst. Publ.,
vol. 33, Cambridge Univ. Press, Cambridge, 1998,
pp. 201–209. MR 1630651
(99e:47027)
 [H]
Kenneth
Hoffman, Banach spaces of analytic functions, PrenticeHall
Series in Modern Analysis, PrenticeHall, Inc., Englewood Cliffs, N. J.,
1962. MR
0133008 (24 #A2844)
 [HRS]
Håkan
Hedenmalm, Stefan
Richter, and Kristian
Seip, Interpolating sequences and invariant subspaces of given
index in the Bergman spaces, J. Reine Angew. Math.
477 (1996), 13–30. MR 1405310
(97i:46044), http://dx.doi.org/10.1515/crll.1996.477.13
 [KM]
Thomas
L. Kriete III and Barbara
D. MacCluer, Meansquare approximation by
polynomials on the unit disk, Trans. Amer.
Math. Soc. 322 (1990), no. 1, 1–34. MR 948193
(91b:30119), http://dx.doi.org/10.1090/S0002994719900948193X
 [M]
Thomas
L. Miller, Some subnormal operators not in 𝐴₂,
J. Funct. Anal. 82 (1989), no. 2, 296–302. MR 987295
(90c:47040), http://dx.doi.org/10.1016/00221236(89)900724
 [OT]
Robert
F. Olin and James
E. Thomson, Some index theorems for subnormal operators, J.
Operator Theory 3 (1980), no. 1, 115–142. MR 565754
(81f:47031)
 [OY1]
Robert
F. Olin and Liming
Yang, A subnormal operator and its dual, Canad. J. Math.
48 (1996), no. 2, 381–396. MR 1393039
(98j:47055), http://dx.doi.org/10.4153/CJM19960211
 [OY2]
Robert
F. Olin and Li
Ming Yang, The commutant of multiplication by 𝑧 on the
closure of polynomials in 𝐿^{𝑡}(𝜇), J. Funct.
Anal. 134 (1995), no. 2, 297–320. MR 1363802
(97m:47023), http://dx.doi.org/10.1006/jfan.1995.1147
 [T]
James
E. Thomson, Approximation in the mean by polynomials, Ann. of
Math. (2) 133 (1991), no. 3, 477–507. MR 1109351
(93g:47026), http://dx.doi.org/10.2307/2944317
 [TY]
James
E. Thomson and Liming
Yang, Invariant subspaces with the codimension one property in
𝐿^{𝑡}(𝜇), Indiana Univ. Math. J.
44 (1995), no. 4, 1163–1173. MR 1386764
(97c:47036), http://dx.doi.org/10.1512/iumj.1995.44.2023
 [Y]
Li
Ming Yang, Invariant subspaces of the Bergman space and some
subnormal operators in 𝐀₁\sbs𝐀₂,
Michigan Math. J. 42 (1995), no. 2, 301–310. MR 1342492
(96f:47013), http://dx.doi.org/10.1307/mmj/1029005230
 [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
