Generalized weighted composition operators on the Bergman space

• Autorzy: Sharma A. K.
• Języki publikacji: EN

Abstrakty

EN

Let φ and ψ be holomorphic maps on the open unit disk D such that φ (D) ⊂ D and H (D) be the space of holomorphic functions on D. For a non-negative integer n, define a linear operator (…), f ∈ H (D). In this paper, we characterize boundedness and compactness of (…) on the Bergman space A². We also compute the upper and lower bounds of essential norm of this operator on the Bergman space.

Słowa kluczowe

EN

generalized weighted composition operator; Bergman space; Nevanlinna counting function

PL

przestrzeń Bergmana; funkcja zliczania Nevanlinna

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2011
• Tom: Vol. 44, nr 2
• Strony: 359--372
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Sharma A. K.

• School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, J&K, India

Bibliografia

• [1] J. E. Brennan, The integrability of the derivative in conformal mapping, J. London Math. Soc. 18 (1978), 261–272.
• [2] C. C. Cowen, B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press Boca Raton, New York, 1995.
• [3] Z. Cuckovic, R. Zhao, Weighted composition operators on the Bergman space, J. London Math. Soc. 70 (2004), 499–511.
• [4] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746–765.
• [5] F. Forelli, The isometries of Hp spaces, Canad. J. Math. 16 (1964), 721–728.
• [6] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer, New York, 2000.
• [7] R.A. Hibschweiler, N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math. 35 (2005), 843–855.
• [8] D. H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. Math. J. 40 (1985), 85–111.
• [9] B. D. MacCluer, R. Zhao, Essential norms of weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math. 33 (2003), 1437–1458.
• [10] V. Matache, Composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc. 127 (1999), 1483–1491.
• [11] E. Nordgren, P. Rosenthal, F. S. Wintrobe, Invertible composition operators on Hp, J. Funct. Anal. 73 (1987), 324–344.
• [12] S. Ohno, Products of composition and differentiation between Hardy spaces, Bull. Austral. Math. Soc. 73 (2006), 235–243.
• [13] A. Montes-Rodriguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. London. Math. Soc. 61 (2000), 872–884.
• [14] J. H. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), 375–404.
• [15] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, New York, 1993.
• [16] J. H. Shapiro, W. Smith, Hardy spaces that support no compact composition operators, J. Funct. Anal. 205 (2003), 62–89.
• [17] A. Sharma, A. K. Sharma, Carleson measures and a class of generalized Integration operators on the Bergman space, Rocky Mountain J. of Math., to appear.
• [18] R. K. Singh, S. D. Sharma, Composition operators on a functional Hilbert space, Bull. Austral. Math. Soc. 20 (1979), 377–384.
• [19] W. Smith, Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (1996), 2331–2348.
• [20] W. Smith, Brennan’s conjecture for weighted composition operators. Recent advances in operator-related function theory, Contemp. Math. 393, 209–214.
• [21] R. Zhao, Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), 139–150.
• [22] X. Zhu, Products of differentiation, composition and multiplication from Bergman type spaces to Bers type space, Integral Transforms. Spec. Funct. 18 (3) (2007), 223–231.
• [23] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.