## Applied harmonic analysis, frame theory, and operator theory

#### Event Title

On invariant Graph subspaces

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.

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Jul 19th, 5:30 PM Jul 19th, 5:50 PM

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.