Algebras on the disk and doubly commuting multiplication operators
Authors:
Sheldon Axler and Pamela Gorkin
Journal:
Trans. Amer. Math. Soc. 309 (1988), 711723
MSC:
Primary 46J15; Secondary 47B35
MathSciNet review:
961609
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We prove that a bounded analytic function on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by and does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if and are bounded analytic functions on the unit disk such that the commutator (here denotes the operator of multiplication by on the Bergman space of the disk) is compact, then as .
 [1]
Sheldon
Axler, The Bergman space, the Bloch space, and commutators of
multiplication operators, Duke Math. J. 53 (1986),
no. 2, 315–332. MR 850538
(87m:47064), http://dx.doi.org/10.1215/S0012709486053202
 [2]
Sheldon
Axler, SunYung
A. Chang, and Donald
Sarason, Products of Toeplitz operators, Integral Equations
Operator Theory 1 (1978), no. 3, 285–309. MR 511973
(80d:47039), http://dx.doi.org/10.1007/BF01682841
 [3]
Sheldon
Axler and Allen
Shields, Algebras generated by analytic and harmonic
functions, Indiana Univ. Math. J. 36 (1987),
no. 3, 631–638. MR 905614
(88h:46102), http://dx.doi.org/10.1512/iumj.1987.36.36034
 [4]
Arlen
Brown and P.
R. Halmos, Algebraic properties of Toeplitz operators, J.
Reine Angew. Math. 213 (1963/1964), 89–102. MR 0160136
(28 #3350)
 [5]
Paul Edward Budde, Support sets and Gleason parts of , Ph.D. thesis, Univ. of California, Berkeley, 1982.
 [6]
Lennart
Carleson, An interpolation problem for bounded analytic
functions, Amer. J. Math. 80 (1958), 921–930.
MR
0117349 (22 #8129)
 [7]
Lennart
Carleson, Interpolations by bounded analytic functions and the
corona problem, Ann. of Math. (2) 76 (1962),
547–559. MR 0141789
(25 #5186)
 [8]
Sun
Yung A. Chang, A characterization of Douglas subalgebras, Acta
Math. 137 (1976), no. 2, 82–89. MR 0428044
(55 #1074a)
 [9]
John
B. Conway, Subnormal operators, Research Notes in Mathematics,
vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.London,
1981. MR
634507 (83i:47030)
 [10]
Theodore
W. Gamelin, Uniform algebras, PrenticeHall, Inc., Englewood
Cliffs, N. J., 1969. MR 0410387
(53 #14137)
 [11]
John
B. Garnett, Bounded analytic functions, Pure and Applied
Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich,
Publishers], New YorkLondon, 1981. MR 628971
(83g:30037)
 [12]
Håkan
Hedenmalm, Thin interpolating sequences and three
algebras of bounded functions, Proc. Amer.
Math. Soc. 99 (1987), no. 3, 489–495. MR 875386
(88c:46065), http://dx.doi.org/10.1090/S00029939198708753868
 [13]
Kenneth
Hoffman, Bounded analytic functions and Gleason parts, Ann. of
Math. (2) 86 (1967), 74–111. MR 0215102
(35 #5945)
 [14]
Keiji Izuchi, Bloch functions and Hankel operators on Bergman spaces in several variables, preprint.
 [15]
Donald
E. Marshall, Subalgebras of 𝐿^{∞} containing
𝐻^{∞}, Acta Math. 137 (1976),
no. 2, 91–98. MR 0428045
(55 #1074b)
 [16]
G.
McDonald and C.
Sundberg, Toeplitz operators on the disc, Indiana Univ. Math.
J. 28 (1979), no. 4, 595–611. MR 542947
(80h:47034), http://dx.doi.org/10.1512/iumj.1979.28.28042
 [17]
Walter
Rudin, Spaces of type 𝐻^{∞}+𝐶, Ann.
Inst. Fourier (Grenoble) 25 (1975), no. 1, vi,
99–125 (English, with French summary). MR 0377520
(51 #13692)
 [18]
Donald Sarason, Blaschke products in , Linear and Complex Analysis Problem Book, Lecture Notes in Math., vol. 1043, SpringerVerlag, Berlin, 1984.
 [19]
Karel Mattheus Rudolf Stroethoff, Characterizations of the Bloch space and related spaces, Ph.D. thesis, Michigan State Univ., 1987.
 [20]
Carl
Sundberg and Thomas
H. Wolff, Interpolating sequences for
𝑄𝐴_{𝐵}, Trans. Amer.
Math. Soc. 276 (1983), no. 2, 551–581. MR 688962
(84e:30078), http://dx.doi.org/10.1090/S00029947198306889623
 [21]
A.
L. Vol′berg, Two remarks concerning the theorem of S. Axler,
S.Y. A. Chang and D. Sarason, J. Operator Theory 7
(1982), no. 2, 209–218. MR 658609
(84h:47038a)
 [22]
Dechao Zheng, Hankel operators and Toeplitz operators on the Bergman space, preprint.
 [1]
 Sheldon Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53 (1986), 305332. MR 850538 (87m:47064)
 [2]
 Sheldon Axler, SunYung A. Chang, and Donald Sarason, Products of Toeplitz operators, Integral Equations Operator Theory 1 (1978), 285309. MR 511973 (80d:47039)
 [3]
 Sheldon Axler and Allen Shields, Algebras generated by analytic and harmonic functions, Indiana Univ. Math. J. 36 (1987), 631638. MR 905614 (88h:46102)
 [4]
 Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1964), 89102. MR 0160136 (28:3350)
 [5]
 Paul Edward Budde, Support sets and Gleason parts of , Ph.D. thesis, Univ. of California, Berkeley, 1982.
 [6]
 Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921930. MR 0117349 (22:8129)
 [7]
 , Interpolations by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547559. MR 0141789 (25:5186)
 [8]
 SunYung A. Chang, A characterization of Douglas subalgebras, Acta Math. 137 (1976), 8189. MR 0428044 (55:1074a)
 [9]
 John B. Conway, Subnormal operators, Pitman, London, 1981. MR 634507 (83i:47030)
 [10]
 Theodore W. Gamelin, Uniform algebras, PrenticeHall, Englewood Cliffs, N. J., 1969. MR 0410387 (53:14137)
 [11]
 John B. Garnett, Bounded analytic functions, Academic Press, New York, 1981. MR 628971 (83g:30037)
 [12]
 Håkan Hedenmalm, Thin interpolating sequences and three bounded algebras of bounded functions, Proc. Amer. Math. Soc. 99 (1987), 489495. MR 875386 (88c:46065)
 [13]
 Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. 86 (1967), 74111. MR 0215102 (35:5945)
 [14]
 Keiji Izuchi, Bloch functions and Hankel operators on Bergman spaces in several variables, preprint.
 [15]
 Donald E. Marshall, Subalgebras of containing , Acta Math. 137 (1976), 9198. MR 0428045 (55:1074b)
 [16]
 G. McDonald and S. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), 595611. MR 542947 (80h:47034)
 [17]
 Walter Rudin, Spaces of type , Ann. Inst. Fourier (Grenoble) 25 (1975), 99125. MR 0377520 (51:13692)
 [18]
 Donald Sarason, Blaschke products in , Linear and Complex Analysis Problem Book, Lecture Notes in Math., vol. 1043, SpringerVerlag, Berlin, 1984.
 [19]
 Karel Mattheus Rudolf Stroethoff, Characterizations of the Bloch space and related spaces, Ph.D. thesis, Michigan State Univ., 1987.
 [20]
 Carl Sundberg and Thomas H. Wolff, Interpolating sequences for , Trans. Amer. Math. Soc. 276 (1983), 551581. MR 688962 (84e:30078)
 [21]
 A. L. Volberg, Two remarks concerning the theorem of S. Axler, S.Y. A. Chang and D. Sarason, J. Operator Theory 7 (1982), 209218. MR 658609 (84h:47038a)
 [22]
 Dechao Zheng, Hankel operators and Toeplitz operators on the Bergman space, preprint.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
46J15,
47B35
Retrieve articles in all journals
with MSC:
46J15,
47B35
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198809616099
PII:
S 00029947(1988)09616099
Article copyright:
© Copyright 1988
American Mathematical Society
