Partially fundamentally reducible operators in Krein spaces

Abstract

A self-adjoint operator $A$ in a Krein space $(\mathcal{K},[\cdot ,\cdot ])$ is called partially fundamentally reducible if there exist a fundamental decomposition $\mathcal{K}={\mathcal{K}}_{+}[\dot{+}]{\mathcal{K}}_{-}$ (which does not reduce $A$) and densely defined symmetric operators $S_{+}$ and $S_{-}$ in the Hilbert spaces $({\mathcal{K}}_{+},[\cdot ,\cdot ])$ and $({\mathcal{K}}_{-},-[\cdot ,\cdot ])$, respectively, such that each $S_{+}$ and $S_{-}$ has defect numbers $(1,1)$ and the operator $A$ is a self-adjoint extension of $S=S_{+}\oplus (-S_{-})$ in the Krein space $(\mathcal{K},[\cdot ,\cdot ])$. The operator $A$ is interpreted as a coupling of operators $S_{+}$ and $-S_{-}$ relative to some boundary triples $(\mathbb{C},{\Gamma }_0^{+},{\Gamma }_1^{+})$ and $(\mathbb{C},{\Gamma }_0^{-},{\Gamma }_1^{-})$. Sufficient conditions for a nonnegative partially fundamentally reducible operator $A$ to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions $m_{+}$ and $m_{-}$ of $S_{+}$ and $S_{-}$ relative to the boundary triples $(\mathbb{C},{\Gamma }_0^{+},{\Gamma }_1^{+})$ and $(\mathbb{C},{\Gamma }_0^{-},{\Gamma }_1^{-})$. Moreover, it is shown that under some asymptotic assumptions on $m_{+}$ and $m_{-}$ all positive self-adjoint extensions of the operator $S$ are similar to self-adjoint operators in a Hilbert space.

45 pages, results presented at the 21st IWOTA 2010 held in Berlin, Germany

References in corpus (1)

• The similarity problem for $J$-nonnegative Sturm-Liouville operators