Let ${\cal B}({\cal H})^n$ be the space  of
$n$-tuples of operators in ${\cal B}({\cal H})$,
the algebra of bounded operators on a Hilbert space 
${\cal H}$. Given a set ${\cal G}$ of maps from
${\cal B}({\cal H})^n$ into itself, which is 
appropriate for our situation, a tuple 
${\bfit T}\in {\cal B}({\cal H})^n$ is said to be 
${\cal G}$-definite (resp., ${\cal G}$-semidefinite) 
if for any $\phi \in {\cal G}$, each term of 
$\phi ({\bfit T})$ is zero (resp., positive), and 
essentially
${\cal G}$-definite (resp., essentially
${\cal G}$-semidefinite)
if 
for any $\phi \in {\cal G}$, the image
of each term of $\phi ({\bfit T})$
by the canonical quotient map of ${\cal B}({\cal H})$ into
${\cal B}({\cal H})/{\cal C}{({\cal H})}$, ${\cal C}({\cal H})$
the algebra of compact operators on ${\cal H}$,
is zero (resp., positive).
We will show that any essentially ${\cal G}$-definite
tuple can be decomposed into a direct sum of a 
${\cal G}$-definite tuple and {\it irreducible} essentially 
${\cal G}$-definite, non  ${\cal G}$-definite tuples, and that 
the parallel statement is also true for essentially 
${\cal G}$-semidefinite tuples.