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Vietnam Journal of Mathematics 36:4(2008) 373-386

Complemented Subspaces in $L^{\infty}(\mathbb D)$

Namita Das

Abstract. In this paper we have shown that the little Bloch space ${\mathcal B}_0$ cannot be complemented in ${\mathcal B}$ and hence $C(\overline\mathbb D)$ cannot be complemented in $L^{\infty}(\mathbb D).$ Further, we have obtained some closed subspaces of $L^{\infty}(\mathbb D)$ that can be complemented in $L^{\infty}(\mathbb D).$ As a consequence of these results we have shown $\{T_{\phi}: \phi\in h^{\infty}(\mathbb D)\}$ can be complemented in ${\mathcal L}(L_a^2)$ and $\{h_{\phi}: \phi\in h^{\infty}(\mathbb D)\}$ cannot be complemented in ${\mathcal L}(L_a^2, (\overline{L_a^2})_0).$ Here $T_{\phi}$ is the Toeplitz operator on the Bergman space $L_a^2, h_{\phi}$ is the little Hankel operator from $L_a^2$ into $(\overline{L_a^2})_0=\{\bar f: f\in L_a^2, f(0)=0\}$ and $h^{\infty}(\mathbb D)$ is the space of bounded harmonic functions on the unit disk $\mathbb D$.

2000 Mathematics Subject Classification: 47B35, 47B38.

Keywords: Complemented subspace, Toeplitz operators, Hankel operators, Bergman space, Bloch space.