matrix of operators properties

Question

Let $T$ a self-adjoint operator bounded from below on $D(T)=H^2(\mathbb R)$ in the ambiant Hilbert space $\mathcal H=L^2(\mathbb R)$, with essential spectrum $[0,+\infty)$ and for discret spectrum some bounded set of negative eigenvalues. Let $L$ the matrix operator defined by $\begin{pmatrix} 0 & 1 \\ T & 0 \end{pmatrix}$ on $\mathcal H \times \mathcal H$. I'm looking for properties of L, and I cannot conclude anything. What are the properties of $L$ : self​-adjoint, bounded, spectrum ?