New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space

• Autorzy: Flavia Colonna
• Języki publikacji: EN

Abstrakty

EN

Let ψ and φ be analytic functions on the open unit disk $\mathbb{D}$ with φ($\mathbb{D}$) ⊆ $\mathbb{D}$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $\mathcal{D}$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations of boundedness for H 1 as well as of compactness for H p, 1 ≤ p < ∞, and $\mathcal{D}$ purely in terms of the symbols ψ and φ.

Słowa kluczowe

EN

Weighted composition operators; Bloch space; Hardy space; Weighted Bergman space; Dirichlet space

Kategorie tematyczne

• 47B38: Operators on function spaces (general)
• 30H20: Bergman spaces, Fock spaces
• 30H30: Bloch spaces
• 30H10: Hardy spaces

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 1
• Strony: 55-73

Daty

wydano

2013-01-01

online

2012-10-24

Twórcy

autor

Flavia Colonna

• George Mason University

Bibliografia

• [1] Anderson J.M., Clunie J., Pommerenke Ch., On Bloch functions and normal functions, J. Reine Angew. Math., 1974, 270, 12–37
• [2] Arazy J., Fisher S.D., The uniqueness of the Dirichlet space among Möbius-invariant Hilbert spaces, Illinois J. Math., 1985, 29(3), 449–462
• [3] Djrbashian A.E., Shamoian F.A., Topics in the Theory of A αp Spaces, Teubner-Texte Math., 105, Teubner, Leipzig, 1988
• [4] Duren P.L., Theory of H p Spaces, Pure Appl. Math., 38, Academic Press, New York-London, 1970
• [5] Duren P., Schuster A., Bergman Spaces, Math. Surveys Monogr., 100, American Mathematical Society, Providence, 2004
• [6] Hedenmalm H., Korenblum B., Zhu K., Theory of Bergman Spaces, Grad. Texts in Math., 199, Springer, New York, 2000 http://dx.doi.org/10.1007/978-1-4612-0497-8
• [7] Li S., Stevic S., Weighted composition operators from Bergman-type spaces into Bloch spaces, Proc. Indian Acad. Sci. Math. Sci., 2007, 117(3), 371–385 http://dx.doi.org/10.1007/s12044-007-0032-y
• [8] Ohno S., Weighted composition operators between H 1 and the Bloch space, Taiwanese J. Math., 2001, 5(3), 555–563
• [9] Ohno S., Stroethoff K., Weighted composition operators from reproducing Hilbert spaces to Bloch spaces, Houston J. Math., 2011, 37(2), 537–558
• [10] Ohno S., Zhao R., Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc., 2001, 63(2), 177–185 http://dx.doi.org/10.1017/S0004972700019250
• [11] Ross W.T., The classical Dirichlet space, In: Recent Advances in Operator-Related Function Theory, Dublin, August 4–6, 2004, Contemp. Math., 393, American Mathematical Society, Providence, 2006, 171–197 http://dx.doi.org/10.1090/conm/393/07378
• [12] Sharma A.K., Kumari R., Weighted composition operators between Bergman and Bloch spaces, Commun. Korean Math. Soc., 2007, 22(3), 373–382 http://dx.doi.org/10.4134/CKMS.2007.22.3.373
• [13] Tjani M., Compact Composition Operators on Some Möbius Invariant Banach Spaces, PhD thesis, Michigan State University, 1996
• [14] Tjani M., Compact composition operators on Besov spaces, Trans. Amer. Math. Soc., 2003, 355(11), 4683–4698 http://dx.doi.org/10.1090/S0002-9947-03-03354-3
• [15] Wulan H., Zheng D., Zhu K., Compact composition operators on BMOA and the Bloch space, Proc. Amer. Math. Soc., 2009, 137(11), 3861–3868 http://dx.doi.org/10.1090/S0002-9939-09-09961-4
• [16] Zhu K.H., Operator Theory in Function Spaces, Monogr. Textbooks Pure Appl. Math., 139, Marcel Dekker, New York, 1990

Identyfikatory

DOI

10.2478/s11533-012-0097-4