#### Event Title

### On invariant Graph subspaces

#### Location

Crow 204

#### Start Date

7-19-2016 5:30 PM

#### End Date

19-7-2016 5:50 PM

#### Description

For unbounded \(2\times 2\) block operator matrices \(B=\begin{pmatrix}A_0&W1\\W_0&A_1\end{pmatrix}\) we investigate the relation between pairs of reducing graph subspaces, solutions to the Riccati equation \[A_1X-XA_0-XW_1X+W_0=0\] and block diagonalization of the operator \(B\). Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrix \(B\). As an application, we provide a way for block diagonalization of a massless two dimensional graphene Hamiltonian. \vspace{6pt} This talk is based on joint work with K.~A.~Makarov and A.~Seelmann.

On invariant Graph subspaces

Crow 204

For unbounded \(2\times 2\) block operator matrices \(B=\begin{pmatrix}A_0&W1\\W_0&A_1\end{pmatrix}\) we investigate the relation between pairs of reducing graph subspaces, solutions to the Riccati equation \[A_1X-XA_0-XW_1X+W_0=0\] and block diagonalization of the operator \(B\). Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrix \(B\). As an application, we provide a way for block diagonalization of a massless two dimensional graphene Hamiltonian. \vspace{6pt} This talk is based on joint work with K.~A.~Makarov and A.~Seelmann.