  Perfect dyadic operators were first introduced in \cite{AHMTT}, where a local
$T(b)$ theorem was proved for such operators. In \cite{AY} it was shown that
for every singular integral operator $T$ with locally bounded kernel on
$\mathbb{R}^n \times \mathbb{R}^n$ there exists a perfect dyadic operator
$\mathbb{T}$ such that $T -\mathbb{T}$ is bounded on $L^p (dx)$ for all
$1<p<\infty$.
  In this paper we show a decomposition of perfect dyadic operators on real
line into four well known operators: two selfadjoint operators, paraproduct and
its adjoint. Based on this decomposition we prove a sharp weighted version of
the $T(1)$ theorem for such operators, which implies $A_2$ conjecture for such
operators with constant which only depends on $\|T(1)\|_{BMO^d}$,
$\|T^*(1)\|_{BMO^d}$ and the constant in testing conditions for $T$. Moreover,
the constant depends on these parameters at most linearly. In this paper we
also obtain sufficient conditions for the two weight boundedness for a perfect
dyadic operator and simplify these conditions under additional assumptions that
weights are in the Muckenhoupt class $A_\infty^d$.
