  The restricted numerical range $W_R(A)$ of an operator $A$ acting on a
$D$-dimensional Hilbert space is defined as a set of all possible expectation
values of this operator among pure states which belong to a certain subset $R$
of the of set of pure quantum states of dimension $D$. One considers for
instance the set of real states, or in the case of composite spaces, the set of
product states and the set of maximally entangled states. Combining the
operator theory with a probabilistic approach we introduce the restricted
numerical shadow of $A$ -- a normalized probability distribution on the complex
plane supported in $W_R(A)$. Its value at point $z \in {\mathbbm C}$ is equal
to the probability that the expectation value $< \psi|A|\psi>$ is equal to $z$,
where $|\psi>$ represents a random quantum state in subset $R$ distributed
according to the natural measure on this set, induced by the unitarily
invariant Fubini--Study measure. Studying restricted shadows of operators of a
fixed size $D=N_A N_B$ we analyse the geometry of sets of separable and
maximally entangled states of the $N_A \times N_B$ composite quantum system.
Investigating trajectories formed by evolving quantum states projected into the
plane of the shadow we study the dynamics of quantum entanglement. A similar
analysis extended for operators on $D=2^3$ dimensional Hilbert space allows us
to investigate the structure of the orbits of $GHZ$ and $W$ quantum states of a
three--qubit system.
