Abstract:

In this paper, we study the operator ${\cal M}=\left[\begin{array}{ccc} 0& I\\ -A& B \end{array} \right]$ which is associated with the second order differential equation $\ddot{z}(t)-B\dot{z}(t)+Az(t)=0$ in a Hilbert space, where $A$ is a self-adjoint and uniformly positive linear operator, $B$ is accretive. We prove that ${\cal M}$ is a boundedly invertible closed operator and $\overline{{\cal M}|_{H_{1}\times H_{1}}}$=${\cal M}$ where $H_{1}={\cal D}(A)$ with the norm $\|x \|_{H_{1}}=\|Ax\|$. And we characterize the spectral distribution of the operator ${\cal M}$ by using the quadratic numerical range of the block operator matrix ${\cal M}|_{H_{1}\times H_{1}}$.

Key words: Unbounded operator, Block operator matrices, Spectrum, Quadratic numerical range