## Taiwanese Journal of Mathematics

### INTEGRAL OPERATORS ON A SUBSPACE OF HOLOMORPHIC FUNCTIONS ON THE DISC

Takahiko Nakazi

#### Abstract

Let $H(D)$ be an algebra of all holomorphic functions on the open unit disc $D$ and $X$ a subspace of $H(D)$. When $g$ is a function in $H(D)$, put $$J_g(f)(z) = \int^z_0 f(\zeta)g^\prime(\zeta)d\zeta~{\rm and}~I_g(f)(z) = \int^z_0 f^\prime(\zeta)g(\zeta)d\zeta \quad (z \in D)$$ for $f$ in $X$. In this paper, we study $J[X] = \{g \in H(D)~;~J_g(f) \in X$ for all $f$ in $X \}$ and $I[X] = \{g \in H(D)~;~I_g(f) \in X$ for all $f$ in $X \}$. We apply the results to concrete spaces. For example, we study $J[X]$ and $I[X]$ when $X$ is a weighted Bloch space, a Hardy space or a Privalov space.

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 2 (2008), 389-400.

Dates
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.twjm/1500574162

Digital Object Identifier
doi:10.11650/twjm/1500574162

Mathematical Reviews number (MathSciNet)
MR2402123

Zentralblatt MATH identifier
1159.30336

#### Citation

Nakazi, Takahiko. INTEGRAL OPERATORS ON A SUBSPACE OF HOLOMORPHIC FUNCTIONS ON THE DISC. Taiwanese J. Math. 12 (2008), no. 2, 389--400. doi:10.11650/twjm/1500574162. https://projecteuclid.org/euclid.twjm/1500574162

#### References

• A. Aleman and J. A. Cima, An integral operator on $H^p$ and Hardy's inequality, J. Analyse Math., 85 (2001), 157-176.
• A. Aleman and A. G. Siskakis, An inlegral operator on $H^p$, Complex Variables Theory Appl., 28 (1995), 149-158.
• P. L. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
• T. W. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.
• W. K. Hayman, On the characterization of functions meromorphic in the unit disk and their integrals, Acta Math., 112 (1964), 181-214.
• H. Helson, Large analytic functions, In: Analysis and Partial Differential Equations, a collection of papers dedicated to Mischa Cotlar, ed. Cora Sadosky, Dekker, 1990, pp. 217-220.
• N. Yanagihara, On a class of functions and their integrals, Proc. London Math. Soc. $($Ser. 3$)$, 25 (1972), 550-576.
• R. Yoneda, Integration operators on weighted Bloch spaces, Nihonkai Math. J., 12 (2001), 123-133.
• R. Yoneda, Multiplication operators, integration operators and companion operators on weighted Bloch spaces, Hokkaido Math. J., 34 (2005), 135-147.