UNBOUNDED SELF-ADJOINT OPERATOR ΠK SPACE

如果A是Πsubsub空间上的自共轭算子，由文[1]可知存在空间昨一个标准分解 \[{\Pi _k} = N \oplus \{ Z + {Z^*}\} \oplus P\] 在此分解下，A有三角模型\[A = \{ S,{A_N},{A_p},F,G,Q\} \].利用三角模型，我们直接证明了 定理1设A是\[{\Pi _k}\]上的-共轭算子，n是任何自然数，那末\[{A^n}\]也是自共轭算子. 定理2设A是\[{A^n}\]上的自共轭算子，那末对所有的\[{A^n}(n = 1,2,...)\]，存在一个公共 的标准分解，在此分解下 \[\begin{gathered} {A^n} = \{ {S^n},A_N^n,A_P^n,\sum\limits_{i = 0}^{n - 1} {{S^i}} FA_N^{n - 1 - i},\sum\limits_{i = 0}^{n - 1} {{S^i}GA_P^{n - 1 - i}} , \hfill \ \sum\limits_{i = 0}^{n - 1} {{S^i}} Q{S^{*n - 1 - i}} - \sum\limits_{i + j + k = n - 2} {{S^i}(FA_N^j{F^*} + GA_P^j{G^*}){S^{*k}}} \} \hfill \\ \end{gathered} \] 定理3 设A是瓜空间上的自共轭算子，\[\sigma (A) \subset [0,\infty ),0 \notin {\sigma _P}(A),\]，那末存在唯 一的自共轭算子A1，满足\[A_1^n = A,\sigma ({A_1}) \subset [0,\infty )\] 其次，我们研究了谱系在临界点附近的性状.记临界点全体为\[C(A)\]).对 \[{\lambda _0} \in C(A)\]记S与入0相应的最高阶根向量的阶数为\[r({\lambda _0})\] 定理4设A是\[{\Pi _k}\]空间上的无界自共轭算子，\[C(A) \cap ({\mu _1},{\nu _1}) = \{ {\lambda _0}\} \],那末以下四 个命题等价： (i）\[\mathop {sup}\limits_{\mu ,\nu } \{ \left\| {{E_{\mu \nu }}} \right\||{\lambda _0} \in (\mu ,\nu ) \subset ({\mu _1},{\nu _1})\} < \infty \] (ii）\[{\mu ^{{\text{1}}}}...,{\mu ^{{{\text{k}}_{\text{0}}}}}\]是全有限的测度； （iii）\[s - \lim {\kern 1pt} {\kern 1pt} {\kern 1pt} {E_{\mu \nu }}\]存在； （iv)A与\[{\lambda _0}\]相应的根子空间\[{\Phi _{{\lambda _0}}}\]非退化；这里\[{\mu ^{{\text{1}}}}...,{\mu ^{{{\text{k}}_{\text{0}}}}}\]是由\[{A_P}\]与G导出的测度. 定通5 设A是\[{\Pi _k}\]上自共轭算子，\[{\lambda _0} \in C(A),r({\lambda _0}) = n\]，那么 （i）\[{E_{\mu \nu }}\]在\[{{\lambda _0}}\]处的奇性次数不超过2n， （ii）\[s - \mathop {\lim }\limits_{\varepsilon \to 0} \int_{[{M_1},{\lambda _0} - \varepsilon )} {(t - {\lambda _0}} {)^{2n}}d{E_t},s - \mathop {\lim }\limits_{\varepsilon \to 0} \int_{[{\lambda _0} + \varepsilon ,{M_2})} {(t - {\lambda _0}} {)^{2n}}d{E_t},\]存在。这里\[{M_1},{M_2}\]满足\[[{M_1},{M_2}] \cap C(A) = \{ {\lambda _0}\} \] 定理6 设A是\[{\Pi _k}\]上的自共轭算子，临界点集\[C(A) = \{ {\lambda _1},...,{\lambda _l},{\lambda _{l + 1}},{\overline \lambda _{l + 1}},...,{\lambda _{l + p}},{\overline \lambda _{l + p}},\]，这里\[\operatorname{Im} {\lambda _v} = 0(1 \leqslant \nu \leqslant l),r({\lambda _\nu }) = {n_\nu }\]那么有 \[{(\lambda - A)^{ - 1}} = \int_{ - \infty }^\infty {K(\lambda ,t)d{E_t}} + \sum\limits_{\nu = 1}^l {\sum\limits_{i = 1}^{2{n_\nu } + 1} {\frac{{{B_{\nu i}}}}{{{{(\lambda - {\lambda _\nu })}^i}}}} } + \sum\limits_{\nu = l + 1}^{l + p} {\sum\limits_{i = 1}^{{n_\nu }} {[\frac{{{B_{\nu i}}}}{{{{(\lambda - {\lambda _\nu })}^i}}}} } + \frac{{B_{\nu i}^ + }}{{{{(\lambda - {{\overline \lambda }_v})}^i}}}]\] 这里 \[K(\lambda ,t) = \frac{1}{{\lambda - t}} - \sum\limits_{v = 1}^l {\delta (t - {\lambda _v}} )\sum\limits_{i = 1}^{2{n_v}} {\frac{{{{(t - {\lambda _v})}^{i - 1}}}}{{{{(\lambda - {\lambda _v})}^i}}}} ,\delta \lambda {\text{ = }}\left\{ \begin{gathered} {\text{1}}{\text{|}}\lambda {\text{| < }}\delta \hfill \ {\text{0}}{\text{|}}\lambda {\text{|}} \geqslant \delta \hfill \\ \end{gathered} \right.\] \[0 < \delta < \mathop {\min }\limits_\begin{subarray}{l} 1 \leqslant \mu ,v \leqslant l \\ {\lambda _\mu } \ne {\lambda _v} \end{subarray} |{\lambda _\mu } - {\lambda _v}|\].对\[1 \leqslant v \leqslant l\]，\[{B_{vi}}\]是\[{\Pi _k}\]上的有界自共轭算子，而当\[l + 1 \leqslant v \leqslant l + p\]时，\[{B_{vi}} = {({\lambda _\mu } - S)^{i - 1}}{P_{\lambda v}}\]是以与\[{{\lambda _v}}\]相应的根子空间为值域的某些平行投影. 定理7 在定理6的条件下，有 \[\begin{gathered} {\text{f}}(A) = \int_{ - \infty }^\infty {[f(t) - \sum\limits_{v = 1}^l {\delta (t - {\lambda _v}} } )\sum\limits_{i = 0}^{2{n_v} - 1} {\frac{{{f^{(i)}}({\lambda _v})}}{{i!}}} (t - {\lambda _v})d{E_t} \hfill \ {\text{ + }}\sum\limits_{{\text{v = 1}}}^{\text{l}} {\sum\limits_{i = 0}^{2{n_v}} {\frac{{{f^{(i)}}({\lambda _0})}}{{i!}}} } {B_v} + \sum\limits_{v = l + 1}^{l + p} {\sum\limits_{i = 0}^{{n_v} - 1} {[\frac{{{f^{(i)}}({\lambda _v})}}{{i!}}} } {B_{vi}} + \frac{{{f^{(i)}}({{\overline \lambda }_v})}}{{i!}}B_{vi}^ + ] \hfill \\ \end{gathered} \] 这里\[f(\lambda )\]在\[\sigma (A)\]的一个邻域内解析. 为了建立更一般的算子演算，我们引入两个特殊的代数： \[{\Omega _n} = \{ (f,\{ {a_i}\} _{i = 0}^{2n})|f\]为Borel可测函数，\[\{ {a_i}\} \]为一常数}。对\[F = (f,\{ {a_i}\} ) \in {\Omega _n},G = (g,\{ {b_i}\} ) \in {\Omega _n}\]，定义 \[\begin{gathered} \alpha F + \beta G = (\alpha f + \beta G,\{ \alpha {a_i} + \beta {b_i}\} ) \hfill \ F \cdot G = (f \cdot g,\{ \sum\limits_{j = 0}^i {{a_j}} {b_{i - j}}\} ),\overline F = (\overline f ,\{ {\overline a _i}\} ) \hfill \\ \end{gathered} \] 显然\[{\Omega _n}\]是一个交换代数，它的子代数\[{\omega _n}\]定义为 \[{\omega _n} = \{ F = (f,\{ {a_i}\} ) \in {\Omega _n}|\]在0点的一个与F有关的邻域中，成立\[{\text{|f(t) - }}\sum\limits_{i = 0}^{2n} {a{t^i}} | \leqslant {M_F}|t{|^{2n + 1}},{M_F}\]与F有关} 定义 设A是\[{\Pi _k}\]上的自共轭算子，C（A）={0}，r(0)=n,对\[F = (f,\{ {a_i}\} ) \in {\omega _n}\]，定义 \[\begin{gathered} FA{\text{ = }}\int_{{\text{ - }}\infty }^\infty {|f(t) - \sum\limits_{i = 0}^{2n} {{a_i}} } {t^i}{|^2}d{E_t} + \sum\limits_{i = 0}^{2n} {{a_i}} {A^i} \hfill \ DF(A)) = D({A^{2n}}) \cap \{ x \in {\Pi _k}\int_{{\text{ - }}\infty }^\infty {|f(t) - \sum\limits_{i = 0}^{2n} {{a_i}} } {t^i}{|^2}d{\left\| {{E_t}x} \right\|^2} < \infty \hfill \\ \end{gathered} \] 如果f解析，\[F = (f,\{ \frac{{{f^{(i)}}(0)}}{{i!}}\} )\]，那么可得F（A）=f（A）。 定理8 设A是有界自共轭算子，C（A）={0}，r（0）=n，\[G \in {\omega _n}\]，那么 \[\begin{gathered} \overline F (A) = {[F(A)]^ + },(\alpha F + \beta G)(A) = \alpha F(A) + \beta G(A) \hfill \ (FG)(A) = F(A)G(A). \hfill \\ \end{gathered} \] 定理9 设A是\[{\Pi _k}\]上的自共轭算子，C（A）={0}，r（0）=n，\[{F_1} = ({f_1},\{ {a_i}\} ) \in {\Omega _n}\]，\[{F_2} = ({f_2},\{ {a_i}\} ) \in {\omega _n},{f_1},{f_2}\]在\[( - \infty ,\infty )\]连续，在\[\sigma (A)\]上恒等，那么\[{F_1}(A) = {F_2}(A)\]。 定理10 设A是\[{\Pi _k}\]上自共轭算子C（A）={0}，r（0）=n，\[F = (f,\{ {a_i}\} ) \in {\Omega _n}\]f是连续函数，那么\[\sigma (F(A)) = \{ f(t)|t \in \sigma (A)\} \]。 在定理11中，我们建立了F（A）的三角模型并由此证明当\[F = \overline F \]时，\[C(F(A)) = \{ f(t)|t \in C(A)\} \] 定理12 设A施可析\[{\Pi _k}\]空间上的自共轭算子，C（A）={0}，r（0）=n，与0相应的根子空间非退化，T是稠定闭算子，那么\[T \in {\{ A\} ^{'}}\]的充要条件是存在\[F \in {\Omega _n}\]，使T=F（A）。这里\[{\{ A\} ^{'}} = \{ T|\]对满足\[BA \subset AB\]的有界算子B，均有\[BT \subset TB\]}