The numerical ranges of the generalized quadratic operators

We investigate the generalized quadratic operator defined by $$T=\left(\begin{array}{cc}a{I}_H& A\\ c{A}^{\ast }& b{I}_K\end{array}\right),$$ where $H$ and $K$ are Hilbert spaces, $A:K\to H$ is a bounded linear operator, $I_H$ and $I_K$ denote the identity operators on $H$ and $K$, respectively, and $a,b,c$ are complex numbers. It is shown that $T$ attains its norm if and only if $A$ attains its norm. Furthermore, a complete characterization of the numerical range of $T$ is provided by a new approach.