Let ${\cal B(H)}^n$ be the algebra of all $n$-tuples of bounded linear 
operators on a separable Hilbert space ${\cal H}$, and ${\cal G}$ 
a set of maps on ${\cal B(H)}^n$ belong to an appropriate class. 
Then any $n$-tuple ${\bmit T}$ can be decomposed into the direct sum 
${\bmit T}_0 \oplus {\bmit T}^{'}$ of the maximum ${\cal G}$-definite 
(respectively, ${\cal G}$-semidefinite) part ${\bmit T}_0$ and the 
completely non ${\cal G}$-definite (resp., non ${\cal G}$-semidefinite) 
part ${\bmit T}^{'}$. It follows that any bounded operator $T$ has 
the maximum $k$-hyponormal part for any positive integer $k$, and so, 
it can be decomposed into the direct sum $T=T_0 \oplus T_1 \oplus T_2 
\oplus \cdots \oplus T_s$ of the completely non hyponormal part $T_0$, 
the $k$-hyponormal but non $(k+1)$-hyponormal part $T_k \ (1\le k<\infty)$ 
and the maximum subnormal part $T_s$.