  Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H},
\langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x,
y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H},$ induces a seminorm
${\|\cdot\|}_A$ on $\mathcal{H}$. Let ${\|T\|}_A,\ w_A(T),$ and $c_A(T)$ denote
the $A$-operator seminorm, the $A$-numerical radius, and the $A$-Crawford
number of an operator $T$ in the semi-Hilbertian space $\big(\mathcal{H},
{\|\cdot\|}_A\big)$, respectively. In this paper, we present some seminorm
inequalities and equalities for semi-Hilbertian space operators. More
precisely, we give some necessary and sufficient conditions for two orthogonal
semi-Hilbertian operators satisfy Pythagoras' equality. In addition, we derive
new upper and lower bounds for the numerical radius of operators in
semi-Hilbertian spaces. In particular, we show that \begin{align*} \frac{1}{16}
{\|TT^{\sharp_{A}} + T^{\sharp_{A}}T\|}^{2}_{A} +
\frac{1}{16}c_{A}\Big(\big(T^2 + (T^{\sharp_{A}})^2\big)^2\Big) \leq w^4_{A}(T)
\leq \frac{1}{8} {\|TT^{\sharp_{A}} + T^{\sharp_{A}}T\|}^{2}_{A} +
\frac{1}{2}w^2_{A}(T^2), \end{align*} where $T^{\sharp_A}$ is a distinguished
$A$-adjoint operator of $T$. Some applications of our inequalities are also
provided.
