Banach-valued Bloch-type functions on the unit ball of a Hilbert space and weak spaces of Bloch-type

Yıl 2023, Cilt: 6 Sayı: 1, 6 - 21, 15.03.2023

Thai Quang

https://doi.org/10.33205/cma.1243686

Cited By: 1

Öz

In this article, we study the space $\mathcal B_\mu(B_X,Y)$ of $Y$-valued Bloch-type functions on the unit ball $B_X$ of an infinite dimensional Hilbert space $X$ with $\mu$ is a normal weight on $B_X$ and $Y$ is a Banach space. We also investigate the characterizations of the space $\mathcal{WB}_\mu(B_X)$ of $Y$-valued, locally bounded, weakly holomorphic functions associated with the Bloch-type space $\mathcal B_\mu(B_X)$ of scalar-valued functions in the sense that $f\in \mathcal{WB}_\mu(B_X)$ if $w\circ f \in \mathcal B_\mu(B_X)$ for every $w \in \mathcal W,$ a separating subspace of the dual $Y'$ of $Y.$

Anahtar Kelimeler

Operators on Hilbert spaces, Bloch spaces, Weak holomorphic

Kaynakça

• O. Blasco, P. Galindo and A. Miralles: Bloch functions on the unit ball of an infinite dimensional Hilbert space, J. Func. Anal., 267 (2014), 1188–1204.
• O. Blasco, P. Galindo, M. Lindström and A. Miralles: Composition operators on the Bloch space of the unit ball of a Hilbert space, Banach J. Math. Anal., 11(2) (2017), 311–334.
• O. Blasco, P. Galindo, M. Lindström and A. Miralles: Interpolating sequences for weighted spaces of analytic functions on the unit ball of a Hilbert space, Revista Mate., Complutense, 32 (2019), 115–139.
• C. Chu, H. Hamada, T. Honda and G. Kohr: Bloch functions on bounded symmetric domains, J. Funct. Anal., 272(6) (2017), 2412–2441.
• K. T. Hahn: Holomorphic mappings of the hyperbolic space into the complex Euclidean space and the Bloch theorem, Canad. J. Math., 27 (1975), 446–458.
• H. Hamada: Bloch-type spaces and extended Cesàro operators in the unit ball of a complex Banach space, Sci. China Math., 62(4) (2019), 617–628.
• S. G. Krantz: Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math., 1(3) (1983), 193–260.
• J. Mujica: A completeness criterion for indutive limits of Banach spaces, In: Functional Analysis, Holomorphy and Approximation Theory II, pp. 319–329, North-Holland Math. Studies, 86, North-Holland Amsterdam, (1984).
• K. F. Ng: On a theorem of Dixmier, Math. Scand., 29 (1971), 279–280.
• T. T. Quang, L. V. Lam and N. V. Dai: On σ(.;W)-holomorphic functions and theorems of Vitali-type, Complex Anal. and Oper. Theory, 7(1) (2013), 237–259.
• A. L. Shields and D. L. Williams: Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc., 162 (1971), 287–302.
• R. M. Timoney: Bloch functions in several complex variables I, Bull. Lond. Math. Soc., 12 (1980), 241–267.
• R. M. Timoney: Bloch functions in several complex variables II, J. Reine Angew. Math., 319 (1980), 1–22.
• Z. Xu: Bloch Type Spaces on the Unit Ball of a Hilbert Space, Czechoslovak Math. J., 69 (2019), 695–711.
• K. Zhu: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, (2005).