Partially Fundamentally Reducible Operators in Kreĭn Spaces

Abstract

A self-adjoint operator A in a Kreĭn 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 Kreĭn space $${(\mathcal{K}, [\cdot , \cdot])}$$ . The operator A is interpreted as a coupling of operators S + and −S − relative to some boundary triples $${\big(\mathbb{C},\,\Gamma_0^+,\,\Gamma_1^+\big)}$$ and $${\big(\mathbb{C},\,\Gamma_0^-,\,\Gamma_1^-\big)}$$ . 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 $${\big(\mathbb{C},\,\Gamma_0^+,\,\Gamma_1^+\big)}$$ and $${\big(\mathbb{C},\,\Gamma_0^-,\Gamma_1^-\big)}$$ . 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.