  Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded
linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot,
\cdot\rangle\big)$. Given a positive operator $A\in\B(\h)$, and a number
$\lambda\in [0,1]$, a seminorm ${\|\cdot\|}_{(A,\lambda)}$ is defined on the
set $\B_{A^{1/2}}(\h)$ of all operators in $\B(\h)$ having an
$A^{1/2}$-adjoint. The seminorm ${\|\cdot\|}_{(A,\lambda)}$ is a combination of
the sesquilinear form ${\langle \cdot, \cdot\rangle}_A$ and its induced
seminorm ${\|\cdot\|}_A$. A characterization of Birkhoff--James orthogonality
for operators with respect to the discussed seminorm is given. Moving $\lambda$
along the interval $[0,1]$, a wide spectrum of seminorms are obtained, having
the $A$-numerical radius $w_A(\cdot)$ at the beginning (associated with
$\lambda=0$) and the $A$-operator seminorm ${\|\cdot\|}_A$ at the end
(associated with $\lambda=1$). Moreover, if $A=I$ the identity operator, the
classical operator norm and numerical radius are obtained. Therefore, the
results in this paper are significant extensions and generalizations of known
results in this area.
