A Volterra type operator on spaces of analytic functions. (English) Zbl 0955.47029

Siskakis, Aristomenis G.; Zhao, Ruhan

Jarosz, Krzysztof (ed.), Function spaces. Proceedings of the 3rd conference, Edwardsville, IL, USA, May 19-23, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 232, 299-311 (1999).

The article studies properties of Volterra type operators in the functional spaces. There are two parts in the paper.
The first part describes general propositions for linear operators \[ I_g(f)(z)= \int^z_0 f(\zeta) g'(\zeta) d\zeta. \] This operator is acting on a Banach space of analytic functions defined on the unit disc in the complex plane \(\mathbb{C}\).
In the second part the operator is acting on BMOA spaces. The terms for \(I_g\) are found when the operator \(I_g\) is bounded or compact. About the operator \(I_g\) on Hardy spaces \(H^p\) see [A. Aleman and A. G. Siskakis, Complex Variables, Theory Appl. 28, No. 2, 149-158 (1995; Zbl 0837.30024)]. About the operator \(I_g\) on Bergman spaces see [A. Aleman and A. G. Siskakis, Indiana Univ. Math. J. 46, No. 2, 337-356 (1997))].
For the entire collection see [Zbl 0913.00036].

Reviewer: V.N.Karpushkin (Moskva)

MSC:

47G10	Integral operators
47B38	Linear operators on function spaces (general)
46E15	Banach spaces of continuous, differentiable or analytic functions
47B07	Linear operators defined by compactness properties
30D55	\(H^p\)-classes (MSC2000)
46J15	Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Keywords:

bounded operator; compact operator; Volterra type operators; BMOA spaces; Hardy spaces; Bergman spaces

Citations:

Zbl 0837.30024