STABILITY OF THE DEFICIENCY INDICES OF SYMMETRIC OPERATORS UNDER SELF-ADJOINT PERTURBATIONS

[摘要] Let $S$ and $T$ be symmetric unbounded operators. Denote by $overline{S+T}$ the closure of the symmetric operator $S+T$. In general, the deficiency indices of $overline{S+T}$ are not determined by the deficiency indices of $S$ and $T$. The paper studies some sufficient conditions for the stability of the deficiency indices of a symmetric operator $S$ under self-adjoint perturbations $T$. One can associate with $S$ the largest closed $^*$-derivation $delta_{S}$ implemented by $S$. We prove that if the unitary operators $exp(i tT)$, for $tinmathbb{R}$, belong to the domain of $delta_{S}$ and $delta_{S}(exp(i tT))ightarrow0$ in the strong operator topology as $tightarrow0$, then the deficiency indices of $S$ and $overline{S+T}$ coincide. In particular, this holds if $S$ and $exp(i tT)$ commute or satisfy the infinitesimal Weyl relation.We also study the case when $S$ and $T$ anticommute: $exp(-i tT)Ssubseteq Sexp(i tT)$, for $tinmathbb{R}$. We show that if the deficiency indices of $S$ are equal, or if the group ${exp(i tT):tinmathbb{R}}$ of unitary operators has no stationary points in the deficiency space of $S$, then $S$ has a self-adjoint extension which anticommutes with $T$, the operator $S+T$ is closed and the deficiency indices of $S$ and $S+T$ coincide.AMS 2000 Mathematics subject classification: Primary 47B25

[学科分类] 数学（综合）

[关键词] symmetric operator;self-adjoint operator;deficiency indices