zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Weighted composition operators on BMOA. (English) Zbl 1163.47018
The author completely characterizes the boundedness and compactness of the weighted composition operator $W_{\psi,\varphi}f=\psi (f\circ \varphi)$ acting on $BMOA$ and $VMOA$ of the unit disc. The results extend and unify those known for the cases $\varphi(z)=z$ and $\psi(z)=1$ corresponding to the multiplication operator $M_\psi$ [see {\it S. Janson}, Ark. Mat. 14, 189--196 (1976; Zbl 0341.43005) and {\it D. A.\thinspace Stegenga}, Am. J. Math. 98, 573--589 (1976; Zbl 0335.47018)] and the composition operator $C_\varphi$ [see {\it P. S.\thinspace Bourdon, J. A.\thinspace Cima} and {\it A. L.\thinspace Matheson}, Trans. Am. Math. Soc. 351, No. 6, 2183--2196 (1999; Zbl 0920.47029) and {\it W. Smith}, Proc. Am. Math. Soc. 127, No. 9, 2715--2725 (1999; Zbl 0921.47025)]. The boundedness of $W_{\psi,\varphi}$ can be described by the facts that the two quantities $\alpha(\psi,\varphi,a)=\vert \psi(a)\vert \Vert \sigma_{\varphi(a)}\circ \varphi\circ \sigma_a\Vert _{H^2}$ and $\beta(\psi,\varphi,a)= (\log\frac{2}{1-\vert \varphi(a)\vert ^2})\Vert \psi\circ \sigma_a-\psi(a)\Vert _{H^2}$, where $\sigma_a$ stands for the Möbious transform mapping $\sigma(0)=a$, are bounded for $\vert a\vert <1$. The proof is based on a weighted version of the Littlewood subordination principle. The author also studies the case $VMOA$ and provides an asymptotic estimate for the essential norm of $W_{\psi,\varphi}$, which seems to be new even in the case of multiplication and composition operators

47B33Composition operators
47B38Operators on function spaces (general)
30H05Bounded analytic functions
30D50Blaschke products, etc. (MSC2000)
46E15Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI Link
[1] J. Arazy, S. D Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110--145. · Zbl 0566.30042
[2] A. Baernstein II, Analytic functions of bounded mean oscillation, in: D. A. Brannan and J. G. Clunie (eds.), Aspects of Contemporary Complex Analysis, Academic Press, London, 1980, pp. 3--36. · Zbl 0492.30026
[3] P. S. Bourdon, J. A. Cima and A. L. Matheson, Compact composition operators on BMOA, Trans. Amer. Math. Soc. 351 (1999), 2183--2196. · Zbl 0920.47029 · doi:10.1090/S0002-9947-99-02387-9
[4] J. A. Cima and A. L. Matheson, Weakly compact composition operators on VMO, Rocky Mountain J. Math. 32 (2002), 937--951. · Zbl 1048.47015 · doi:10.1216/rmjm/1034968424
[5] M. D. Contreras and A.G. Hernández-Díaz, Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263 (2001), 224--233. · Zbl 1026.47016 · doi:10.1006/jmaa.2001.7610
[6] M. D. Contreras and A.G. Hernández-Díaz, Weighted composition operators between different Hardy spaces, Integral Equations Operator Theory 46 (2003), 165--188.
[7] C. C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995. · Zbl 0873.47017
[8] Ž. Čučković and R. Zhao, Weighted composition operators on the Bergman space, J. London Math. Soc. 70 (2004), 499--511. · Zbl 1069.47023 · doi:10.1112/S0024610704005605
[9] Ž. Čučković and R. Zhao, Weighted composition operators between different weighted Bergman and Hardy spaces, Illinois J. Math. 51 (2007), 479--498. · Zbl 1147.47021
[10] R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces: Function Spaces, Chapman &amp; Hall/CRC, Boca Raton, 2003. · Zbl 1011.46001
[11] J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. · Zbl 0469.30024
[12] D. Girela, Analytic functions of bounded mean oscillation, in: R. Aulaskari (ed.), Complex Function Spaces (Mekrijärvi, 1999), Univ. Joensuu Dept. Math. Rep. Ser., 4, Univ. Joensuu, Joensuu, 2001, pp. 61--170. · Zbl 0981.30026
[13] S. Janson, On functions with conditions on the mean oscillation, Ark. Math. 14 (1976), 189--196. · Zbl 0341.43005 · doi:10.1007/BF02385834
[14] J. Laitila, Composition operators and vector-valued BMOA, Integral Equations Operator Theory 58 (2007), 487--502. · Zbl 1133.47020 · doi:10.1007/s00020-007-1503-3
[15] M. Lindström, S. Makhmutov and J. Taskinen, The essential norm of a Bloch-to-Qp composition operator, Canad. Math. Bull. 47 (2004), 49--59. · Zbl 1082.47020 · doi:10.4153/CMB-2004-007-6
[16] B. D. MacCluer and R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), 1437--1458. · Zbl 1061.30023 · doi:10.1216/rmjm/1181075473
[17] S. Makhmutov and M. Tjani, Composition operators on some Möbius invariant Banach spaces, Bull. Austral. Math. Soc. 62 (2000), 1--19. · Zbl 0963.47023 · doi:10.1017/S0004972700018426
[18] A. Montes-Rodríguez, The essential norm of a composition operator on Bloch spaces, Pacific J. Math. 188 (1999), 339--351. · Zbl 0932.30034 · doi:10.2140/pjm.1999.188.339
[19] S. Ohno and R. Zhao, Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc. 63 (2001), 177--185. · Zbl 0985.47022 · doi:10.1017/S0004972700019250
[20] J. M. Ortega and J. Fàbrega, Pointwise multipliers and corona type decomposition in BMOA, Ann. Inst. Fourier (Grenoble) 46 (1996), 111--137. · Zbl 0840.32001 · doi:10.5802/aif.1509
[21] J. H. Shapiro, The essential norm of a composition operator, Ann. of Math. 125 (1987), 375--404. · Zbl 0642.47027 · doi:10.2307/1971314
[22] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York 1993. · Zbl 0791.30033
[23] W. Smith, Compactness of composition operators on BMOA, Proc. Amer. Math. Soc. 127 (1999), 2715--2725. · Zbl 0921.47025 · doi:10.1090/S0002-9939-99-04856-X
[24] D. A. Stegenga, Bounded Toeplitz operators on H1 and applications of the duality between H1 and the functions of bounded mean oscillation, Amer. J. Math. 98 (1976), 573--589. · Zbl 0335.47018 · doi:10.2307/2373807
[25] K. Stephenson, Weak subordination and stable classes of meromorphic functions, Trans. Amer. Math. Soc. 262 (1980), 565--577. · Zbl 0445.30029 · doi:10.1090/S0002-9947-1980-0586736-2
[26] M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Thesis, Michigan State University, 1996.
[27] M. Wang and P. Liu, Weighted composition operators between Hardy spaces, Math. Appl. (Wuhan) 16 (2003), 130--135. · Zbl 1061.47025
[28] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990 · Zbl 0706.47019