We show that for an algebraic Riccati equation $ X^*B^{-1}X - T^*X - X^*T = C $, its solutions are given by $X=W+BT$ for some solution $W$ of $X^*B^{-1}X = C + T^*BT$. To generalize this, we give an equivalent condition for $ egin{pmatrix} B & W W^* & A end{pmatrix} ge 0 $ for given positive operators $B$ and $A$, by which it can be regarded as Riccati inequality $ X^*B^{-1}X le A$. As an application, the harmonic mean $B ! C$ is explicitly written even if $B$ and $C$ are noninvertible.

Keywords: Riccati equation, operator matrix, geometric mean and harmonic mean