On the Dirichlet space for finitely connected regions
Author:
Kit Chak Chan
Journal:
Trans. Amer. Math. Soc. 319 (1990), 711728
MSC:
Primary 46E20; Secondary 30H05, 47B38
MathSciNet review:
958885
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Abstract: This paper is devoted to the study of the Dirichlet space for finitely connected regions ; we are particularly interested in the algebra of bounded multiplication operators on this space. Results in different directions are obtained. One direction deals with the structure of closed subspaces invariant under all bounded multiplication operators. In particular, we show that each such subspace contains a bounded function. For regions with circular boundaries we prove that a finite codimensional closed subspace invariant under multiplication by must be invariant under all bounded multiplication operators, and furthermore it is of the form , where is a polynomial with all its roots lying in . Another direction is to study cyclic and noncyclic vectors for the algebra of all bounded multiplication operators. Typical results are: if and is bounded away from zero then is cyclic; on the other hand, if the zero set of the radial limit function of on the boundary has positive logarithmic capacity, then is not cyclic. Also, some other sufficient conditions for a function to be cyclic are given. Lastly, we study transitive operator algebras containing all bounded multiplication operators; we prove that they are dense in the algebra of all bounded operators in the strong operator topology.
 [Arve]
William
B. Arveson, A density theorem for operator algebras, Duke
Math. J. 34 (1967), 635–647. MR 0221293
(36 #4345)
 [AxBo]
Sheldon
Axler and Paul
Bourdon, Finitecodimensional invariant
subspaces of Bergman spaces, Trans. Amer. Math.
Soc. 306 (1988), no. 2, 805–817. MR 933319
(89f:46051), http://dx.doi.org/10.1090/S00029947198809333195
 [AxSh]
Sheldon
Axler and Allen
L. Shields, Univalent multipliers of the Dirichlet space,
Michigan Math. J. 32 (1985), no. 1, 65–80. MR 777302
(86c:30043), http://dx.doi.org/10.1307/mmj/1029003133
 [BeFP]
Hari
Bercovici, Ciprian
Foias, and Carl
Pearcy, Dual algebras with applications to invariant subspaces and
dilation theory, CBMS Regional Conference Series in Mathematics,
vol. 56, Published for the Conference Board of the Mathematical
Sciences, Washington, DC, 1985. MR 787041
(87g:47091)
 [BrSh]
Leon
Brown and Allen
L. Shields, Cyclic vectors in the Dirichlet
space, Trans. Amer. Math. Soc.
285 (1984), no. 1,
269–303. MR
748841 (86d:30079), http://dx.doi.org/10.1090/S00029947198407488410
 [ChPS]
B.
Chevreau, C.
M. Pearcy, and A.
L. Shields, Finitely connected domains 𝐺, representations
of 𝐻^{∞}(𝐺), and invariant subspaces, J.
Operator Theory 6 (1981), no. 2, 375–405. MR 643698
(83j:47004)
 [DuRS]
P.
L. Duren, B.
W. Romberg, and A.
L. Shields, Linear functionals on 𝐻^{𝑝} spaces with
0<𝑝<1, J. Reine Angew. Math. 238
(1969), 32–60. MR 0259579
(41 #4217)
 [Nguy]
Nguyen
Xuan Uy, Removable sets of analytic functions satisfying a
Lipschitz condition, Ark. Mat. 17 (1979), no. 1,
19–27. MR
543500 (81i:30076), http://dx.doi.org/10.1007/BF02385454
 [Shie]
A.
L. Shields, Cyclic vectors in some spaces of analytic
functions, Proc. Roy. Irish Acad. Sect. A 74 (1974),
293–296. Spectral Theory Symposium (Trinity College, Dublin, 1974).
MR
0361836 (50 #14281)
 [Arve]
 W. B. Arveson, A density for operator algebras, Duke Math. J. 34 (1967), 635647. MR 0221293 (36:4345)
 [AxBo]
 S. Axler and P. Bourdon, Finite codimensional invariant subspaces of Bergman spaces, preprint, 1986. MR 933319 (89f:46051)
 [AxSh]
 S. Axler and A. L. Shields, Univalent multipliers of the Dirichlet space, Michigan Math. J. 32 (1985), 6580. MR 777302 (86c:30043)
 [BeFP]
 H. Bercovici, C. Foias and C. M. Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conf. Ser. in Math., no. 56, Amer. Math. Soc., Providence, R.I., 1985. MR 787041 (87g:47091)
 [BrSh]
 L. Brown and A. L. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math. Soc. 285 (1984), 269304. MR 748841 (86d:30079)
 [ChPS]
 B. Chevreau, C. M. Pearcy and A. L. Shields, Finitely connected domains , representations of , and invariant subspaces, J. Operator Theory 6 (1981), 375405. MR 643698 (83j:47004)
 [DuRS]
 P. L. Duren, B. W. Romberg, and A. L. Shields, Linear functionals on spaces with , J. Reine Angew. Math. 288 (1969), 3260. MR 0259579 (41:4217)
 [Nguy]
 Nguyen Xuan Uy, Removable sets of analytic functions satisfying a Lipschitz condition, Ark. Math. 17 (1979), 1927, MR 81i:30076. MR 543500 (81i:30076)
 [Shie]
 A. L. Shields, Cyclic vectors in some spaces of analytic functions, Proc. Roy. Irish Acad. Sect. A 74 (1974), 293296. MR 0361836 (50:14281)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009588854
PII:
S 00029947(1990)09588854
Keywords:
Dirichlet space,
finitely connected region,
multiplication operators,
invariant subspaces,
cyclic vectors,
transitive operator algebras,
logarithmic capacity
Article copyright:
© Copyright 1990 American Mathematical Society
