## Banach Journal of Mathematical Analysis

### Certain distance estimates for operators on the Bergman space

#### Abstract

Let $\mathbb{D}$ be the open unit disk with its boundary $\partial\mathbb{D}$ in the complex plane $\mathbb{C}$ and $dA(z)=\frac{1}{\pi}dx\, dy,$ the normalized area measure on $\mathbb{D}.$ Let $L_{a}^{2}(\mathbb{D}, dA)$ be the Bergman space consisting of analytic functions on $\mathbb{D}$ that are also in $L^2(\mathbb{D}, dA).$ In this paper we obtain certain distance estimates for bounded linear operators defined on the Bergman space.

#### Article information

Source
Banach J. Math. Anal., Volume 8, Number 2 (2014), 193-203.

Dates
First available in Project Euclid: 4 April 2014

https://projecteuclid.org/euclid.bjma/1396640063

Digital Object Identifier
doi:10.15352/bjma/1396640063

Mathematical Reviews number (MathSciNet)
MR3189550

Zentralblatt MATH identifier
1303.47029

#### Citation

Das, Namita; Sahoo, Madhusmita. Certain distance estimates for operators on the Bergman space. Banach J. Math. Anal. 8 (2014), no. 2, 193--203. doi:10.15352/bjma/1396640063. https://projecteuclid.org/euclid.bjma/1396640063

#### References

• A.W. Marshall and I. Olkin, Inequalities: theory of majorization and its application, Academic Press, New York, 1979.
• F. Kittaneh, Norm inequalities for commutators of positive operators and applications, Math. Z. 258 (2008), 845–849.
• G. Pisier, Similarity problems and completely bounded maps, 2nd edition, Lecture Notes in Math. 1618, Springer-Verlag, Berlin, 2001.
• I. Gohberg and S. Goldberg, Basic operator theory, Birkhauser, 1981.
• K. Davidson and S.C. Power, Best approximation in $C^{*}$-algebras, J. Reine Angew. Math. 368 (1986), 43–62.
• K. Zhu, On certain unitary operators and composition operators, Proc. Symp. Pure Math. 51 (1990), 371–385.
• M. Nagisa and S. Wada, Averages of operators and their positivity, Proc. Amer. Math. Soc. 126 (1998), no. 2, 499–506.
• N. Das, The Berezin transform of bounded linear operators, J. Indian Math. Soc. 76 (2009), 47–60.
• P.J. Maher, Some operator inequalities concerning generalized inverses, Illinois J. Math. 34 (1990), 503–514.
• P.J. Maher, Some norm inequalities concerning generalized inverses, Linear Algebra Appl. 174 (1992), 99–110.
• S. Axler and D. Zheng, The Berezin transform on the Toeplitz algebra, Studia Math. 127 (1998), no. 2, 113–136.