Certain distance estimates for operators on the Bergman space

Namita Das; Madhusmita Sahoo

doi:10.15352/bjma/1396640063

2014 Certain distance estimates for operators on the Bergman space

Namita Das, Madhusmita Sahoo

Banach J. Math. Anal. 8(2): 193-203 (2014). DOI: 10.15352/bjma/1396640063

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.

References

1.

A.W. Marshall and I. Olkin, Inequalities: theory of majorization and its application, Academic Press, New York, 1979. MR552278 A.W. Marshall and I. Olkin, Inequalities: theory of majorization and its application, Academic Press, New York, 1979. MR552278

2.

F. Kittaneh, Norm inequalities for commutators of positive operators and applications, Math. Z. 258 (2008), 845–849. MR2369059 10.1007/s00209-007-0201-9 F. Kittaneh, Norm inequalities for commutators of positive operators and applications, Math. Z. 258 (2008), 845–849. MR2369059 10.1007/s00209-007-0201-9

3.

G. Pisier, Similarity problems and completely bounded maps, 2nd edition, Lecture Notes in Math. 1618, Springer-Verlag, Berlin, 2001. MR1818047 G. Pisier, Similarity problems and completely bounded maps, 2nd edition, Lecture Notes in Math. 1618, Springer-Verlag, Berlin, 2001. MR1818047

4.

I. Gohberg and S. Goldberg, Basic operator theory, Birkhauser, 1981. MR632943 I. Gohberg and S. Goldberg, Basic operator theory, Birkhauser, 1981. MR632943

5.

K. Davidson and S.C. Power, Best approximation in $C^{*}$-algebras, J. Reine Angew. Math. 368 (1986), 43–62. MR850614 K. Davidson and S.C. Power, Best approximation in $C^{*}$-algebras, J. Reine Angew. Math. 368 (1986), 43–62. MR850614

6.

K. Zhu, On certain unitary operators and composition operators, Proc. Symp. Pure Math. 51 (1990), 371–385. MR1077459 K. Zhu, On certain unitary operators and composition operators, Proc. Symp. Pure Math. 51 (1990), 371–385. MR1077459

7.

M. Nagisa and S. Wada, Averages of operators and their positivity, Proc. Amer. Math. Soc. 126 (1998), no. 2, 499–506. MR1415335 10.1090/S0002-9939-98-04070-2 M. Nagisa and S. Wada, Averages of operators and their positivity, Proc. Amer. Math. Soc. 126 (1998), no. 2, 499–506. MR1415335 10.1090/S0002-9939-98-04070-2

8.

N. Das, The Berezin transform of bounded linear operators, J. Indian Math. Soc. 76 (2009), 47–60. MR2778858 N. Das, The Berezin transform of bounded linear operators, J. Indian Math. Soc. 76 (2009), 47–60. MR2778858

9.

P.J. Maher, Some operator inequalities concerning generalized inverses, Illinois J. Math. 34 (1990), 503–514. MR1053559 euclid.ijm/1255988167 P.J. Maher, Some operator inequalities concerning generalized inverses, Illinois J. Math. 34 (1990), 503–514. MR1053559 euclid.ijm/1255988167

10.

P.J. Maher, Some norm inequalities concerning generalized inverses, Linear Algebra Appl. 174 (1992), 99–110. MR1176454 10.1016/0024-3795(92)90045-C P.J. Maher, Some norm inequalities concerning generalized inverses, Linear Algebra Appl. 174 (1992), 99–110. MR1176454 10.1016/0024-3795(92)90045-C

11.

S. Axler and D. Zheng, The Berezin transform on the Toeplitz algebra, Studia Math. 127 (1998), no. 2, 113–136. MR1488147 S. Axler and D. Zheng, The Berezin transform on the Toeplitz algebra, Studia Math. 127 (1998), no. 2, 113–136. MR1488147

Citation Download Citation

Namita Das and Madhusmita Sahoo "Certain distance estimates for operators on the Bergman space," Banach Journal of Mathematical Analysis 8(2), 193-203, (2014). https://doi.org/10.15352/bjma/1396640063

Published: 2014

Access the abstract

JOURNAL ARTICLE
11 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY