  If $\left(\h,\langle\cdot,\cdot\rangle\right)$ is a Hilbert space and on it
we consider the sesquilinear form $\langle\,W\cdot,\cdot\rangle$ so-called
$W$-metric, where $W^{*}=W\in\BH$, and $\ker\,W=\{0\}$, then the space
$\left(\h,\langle\,W\cdot,\cdot\rangle\right)$ is called Hilbert space with
$W$-metric or simply $W$-space. In this paper we investigate the dynamic of
frames of subspace on these spaces, where the sense of dynamics refers to the
behavior of frames of subspace in $\h_{W}$ (the completion of
$\left(\h,\langle\,W\cdot,\cdot\rangle\right)$) comparing with $\h$ and vice
versa. This work is based on the study made in \cite{KEFER,GMMM} on frames in
Krein spaces. In a similar way, Casazza and Kutyniok obtained some results in
the context of Hilbert spaces, see \cite{CG}.
  We take tools of theory of $C^{*}$-algebra, and properties of $\BH$, to show
that every Hilbert space with $W$-metric $\h_{W}$ with $0\in\sigma(W)$ has a
decomposition $$\h_{W}=\bigoplus_{n\in\N\cup\{\infty\}}\h_{\psi_{n}}^{W},$$
where $\h_{\psi_{n}}^{W}\simeq \Ele(\sigma(W),x\,d\mu_{n}(x))$ are Krein
spaces, for every $n\in\N\cup\{\infty\}$. Moreover, we investigate the dynamics
of frames of subspace when the self-adjoint operator $W$ is unbounded.
