  We consider the Kalman - Yakubovich - Popov (KYP) inequality \[
\begin{pmatrix} X-A^* XA-C^*C & -A^*X B- C^*D\cr -B^*X A-D^* C & I- B^*X B-D^*D
\end{pmatrix} \ge 0 \] for contractive operator matrices $ \begin{pmatrix}
A&B\cr C &D \end{pmatrix}:\begin{pmatrix}\mathfrak{H}\cr\mathfrak{M}
\end{pmatrix}\to\begin{pmatrix}\mathfrak{H}\cr\mathfrak{N} \end{pmatrix}, $
where $\mathfrak{H},$ $\mathfrak{M}$, and $\mathfrak{N}$ are separable Hilbert
spaces. We restrict ourselves to the solutions $X$ from the operator interval
$[0, I_\mathfrak{H}]$. Several equivalent forms of KYP are obtained. Using the
parametrization of the blocks of contractive operator matrices, the Kre\u{\i}n
shorted operator, and the M\"obius representation of the Schur class
operator-valued function we find several equivalent forms of the KYP
inequality. Properties of solutions are established and it is proved that the
minimal solution of the KYP inequality satisfies the corresponding algebraic
Riccati equation and can be obtained by the iterative procedure with the
special choice of the initial point. In terms of the Kre\u{\i}n shorted
operators a necessary condition and some sufficient conditions for uniqueness
of the solution are established.
