Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces In this article, we establish a new atomic decomposition for $f\in L^2_w\cap H^p_w$, where the decomposition converges in $L^2_w$-norm rather than in the distribution sense. As applications of this decomposition, assuming that $T$ is a linear operator bounded on $L^2_w$ and $0 Keywords:$A_p$weights, atomic decomposition, CalderÃ³n reproducing formula, weighted Hardy spacesCategories:42B25, 42B30 2. CMB 1998 (vol 41 pp. 196) Nakazi, Takahiko  Brown-Halmos type theorems of weighted Toeplitz operators The spectra of the Toeplitz operators on the weighted Hardy space$H^2(Wd\th/2\pi)$and the Hardy space$H^p(d\th/2\pi)$, and the singular integral operators on the Lebesgue space$L^2(d\th/2\pi)\$ are studied. For example, the theorems of Brown-Halmos type and Hartman-Wintner type are studied. Keywords:Toeplitz operator, singular integral, operator, weighted Hardy space, spectrum.Category:47B35