Krein空间中的部分基本可约算子
Partially fundamentally reducible operators in Krein spaces
摘要：
Krein空间$\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$中的自伴算子$A$称为部分基本可约的，如果存在基本分解${\mathcal K} = {\mathcal K}_+ [\dot{+}] {\mathcal K}_-$（其不约化$A$）和分别在希尔伯特空间$\bigl({\mathcal K}_+,[\,\cdot\,,\cdot\,]\bigr)$和$\bigl({\mathcal K}_-,-[\,\cdot\,,\cdot\,]\bigr)$中密集定义的对称算子$S_+$和$S_-$，使得每个$S_+$和$S_-$具有缺陷号$(1,1)$，并且操作符$A$是自Krein空间$\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$中$S =S_+ \oplus (-S_-)$的伴随扩张。算子$A$被解释为算子$S_+$和$-S_-$相对于某些边界三元组$\bigl({\mathbb C},Γ_0^+,Γ_1^+\bigr)$和$\bigl({\mathbb C},Γ_0^-,Γ_1^-\bigr)$的耦合。利用$S_+$和$S_-$相对于边界三元组$\bigl({\mathbb C},Γ_0^+,Γ_1^+\bigr)$和$\bigl({\mathbb C},Γ_0^-,Γ_1^-\bigr)$的Weyl函数$m_+$和$m_-$，给出了Hilbert空间中非负部分基本可约算子$A$与自伴算子相似的充分条件.此外，还证明了在$m_+$和$m_-$的某些渐近假设下，算子$S$的所有正自伴扩张都类似于Hilbert空间中的自伴算子.
英文摘要：
A self-adjoint operator $A$ in a Krein space $\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$ 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 $\bigl({\mathcal K}_+,[\,\cdot\,,\cdot\,]\bigr)$ and $\bigl({\mathcal K}_-,-[\,\cdot\,,\cdot\,]\bigr)$, 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 $\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$. The operator $A$ is interpreted as a coupling of operators $S_+$ and $-S_-$ relative to some boundary triples $\bigl({\mathbb C},Γ_0^+,Γ_1^+\bigr)$ and $\bigl({\mathbb C},Γ_0^-,Γ_1^-\bigr)$. 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 $\bigl({\mathbb C},Γ_0^+,Γ_1^+\bigr)$ and $\bigl({\mathbb C},Γ_0^-,Γ_1^-\bigr)$. 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.
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