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zenaの日記

麻雀と最近放置している不等式と、極希に更新しているスペクトル以外には、ここを見る価値は基本的にありません。また、管理者は気まぐれにしか更新しませんので悪しからず。

スペクトル

メモとしていろんな(有界線形作用素の)スペクトルをあげてみます。(書き途中)
(最初の6つ以外はそんなに知られていないかも・・・。名称が複数ある場合の順番は著者の独断と偏見でつけています。意味はありません。)


 \mbox{Let}\quad X\quad\mbox{be a Banach space}
 \mbox{and} \quad T\in B(X)\quad\mbox{i.e. }T\mbox{ is a bounded linear operator in }\quad X
 \sigma(T)=\{\lambda :\quad T-\lambda \quad\mbox{ is not bijective}\}\mbox{ (spectrum)}
 \sigma_{\rm ess}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not Fredholm operator}\}(=\sigma_{\rm f}(T)\quad)
 \mbox{ \quad\quad\quad(essential spectrum, essential spectrum 3, Fredholm spectrum)}
 \sigma_{\rm disc}(T)=\{\lambda\quad |\quad\lambda\mbox{ is isolated spectrum and }\lambda\mbox{ has finite algebraic multiplicity}\}
 \quad\quad\quad\mbox{ (discrete spectrum)}
 \sigma_{\rm p}(T)=\{\lambda :\quad T-\lambda\mbox{ is not injective}\}\mbox{ (point spectrum)}
 \sigma_{\rm c}(T)=\left\{\lambda :\quad T-\lambda\mbox{ is injective and }(T-\lambda)(X)\quad\mbox{ is dense in}\quad X\mbox{ but it is not surjective}\right\}\mbox{ (continuous spectrum)}
 \sigma_{\rm r}(T)=\{\lambda :\quad T-\lambda\mbox{ is injective but}\quad (T-\lambda)(X)\quad\mbox{ is not dense in}\quad X\}\mbox{ (residual spectrum)}
 \sigma_{\rm ap}(T)=\{\lambda :\quad T-\lambda\mbox{ is not bounded below}\}\mbox{ (approximat point spectrum)}
 \sigma_{\rm su}(T)=\{\lambda :\quad T-\lambda\mbox{ is not surjective}\}(=\sigma_{\rm ad}(T)\quad)
 \mbox{ \quad\quad\quad(surjectivity spectrum, approximate defect spectrum)}
 \sigma_{\rm com}(T)=\{\lambda :\quad (T-\lambda)(X)\mbox{ is not dense}\}(=\sigma_{\rm d}(T)\quad)\mbox{ (compression spectrum, defect spectrum)}
 \sigma_{\rm se}(T)=\{\lambda :\quad T-\lambda\mbox{ is not semiregular}\}(=\sigma_{\rm k}(T)\quad)
 \mbox{ \quad\quad\quad(semiregular spectrum, Kato spectrum, Apostol spectum)}
 \sigma_{\rm es}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not essentially semiregular}\}\mbox{ (essential semiregular spectrum)}
 \sigma_{\rm kt}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not Kato type operator}\}\mbox{ (Kato type spectrum)}
 \sigma_{\rm reg}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not regular}\}\mbox{ (regular spectrum)}
 \sigma_{\rm f}(T)=\sigma(T)\cup_K\{K\subset\rho (T):\quad\mbox{ bounded connected}\}\mbox{ (full spectrum)}
 \sigma_{\rm le}(T)=\left\{\lambda :\quad (T-\lambda)(X)\quad\mbox{ is not closed or}\quad \ker(T-\lambda)\quad\mbox{is infinite dimensional}\}
 \quad\quad\quad\quad\quad\quad\quad\quad=\{\lambda :\quad T-\lambda\quad\mbox{ is not upper semiFredholm operator}\}(=\sigma_{\rm uf}(T)\quad)
 \mbox{ \quad\quad\quad(left essential spectrum, upper semiFredholm spectrum)}
 \sigma_{\rm re}(T)=\{\lambda :\quad X/(T-\lambda)(X)\quad\mbox{ is infinite dimensional}\}
 \quad\quad\quad\quad\quad\quad\quad\quad\quad =\{\lambda :\quad T-\lambda\quad\mbox{ is not lower semiFredholm operator}\}(=\sigma_{\rm lf}(T)\quad)
 \mbox{ \quad\quad\quad(right essential spectrum, lower semiFredholm spectrum)}
 \sigma_{\rm sf}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not semiFredholm operator}\}
 \mbox{ \quad\quad\quad(semiFredholm spectrum, essential spectrum 1)}
 \sigma_{\rm b}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not Browder operator}\}\mbox{ (Browder spectrum, essential spectrum ?)}
 \sigma_{\rm lb}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not lower semi-Browder operator}\}\mbox{ (lower semi-Browder spectrum)}
 \sigma_{\rm ub}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not upper semi-Browder operator}\}\mbox{ (upper semi-Browder spectrum)}
 \sigma_{\rm w}(T)=\{\lambda :\quad T-\lambda\quad\mbox{ is not Weyl operator}\}\mbox{ (Weyl spectrum, essential spectrum 4)}
 \sigma_{\rm wa}(T)=\cap_{K\in K(X)}\sigma_{\rm ap}(T+K)\mbox{ (Weyl approximate point spectrum)}
 \sigma_{\rm ws}(T)=\cap_{K\in K(X)}\sigma_{\rm su}(T+K)\mbox{ (Weyl surjectivity spectrum)
\ \sigma_{\rm per}(T)=\{\lambda\in\sigma(T):\quad |\lambda|=r(T)\} \mbox{(peripheral spectrum)



補足

  • \white K(X)=\{T\in B(X)\quad |\quad T\mbox{ is compact}\}.
  • \white r(T): spectral radius.
  • semiregular operator : hyperkernel がhyperange に含まれる。
  • Browder operator : finite ascent かつ finite descentをもつ作用素
  • Weyl operator : Fredholm indexが0
  • semiFredholm, Weyl spectrumもessential spectrumと呼ぶ事もあります。理由は恐らくTが自己共役作用素の場合にすべて一致する、コンパクト作用素の摂動に対して不変等の性質を持っているため。
  • Browder spectrum もessential spectrumと呼ばれる事もあるが良く分からない。