Banach代数中元素的谱

2022-11-12 15:18 作者:臥煙鈴 0人读过 | 我要投稿

今天这一章呢其实就是把上一题坑给填了

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$(x-%5Clambda%20e)y%3Dy(x-%5Clambda%20e)%3De$

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$x%E7%9A%84%E6%AD%A3%E5%88%99%E7%82%B9%E5%85%A8%E4%BD%93%E8%AE%B0%E4%B8%BA%5Crho(x)%3B$

$%E9%9D%9E%E6%AD%A3%E5%88%99%E7%82%B9%E8%AE%B0%E4%B8%BAx%E7%9A%84%E8%B0%B1%E7%82%B9%EF%BC%8C%E5%85%B6%E5%85%A8%E4%BD%93%E8%AE%B0%E4%BD%9C%5Csigma(x).$

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证明:

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$%5Cforall%20%5Cvarepsilon%3E0%2C%E5%8F%96m%5Cin%5Cmathbb%7BZ%7D_%7Bx%3E0%7D%E4%BD%BF%E5%BE%97%5CVert%20x%5Em%5CVert%5E%7B%5Cfrac%7B1%7D%7Bm%7D%7D%3C%20r%2B%5Cvarepsilon$

$%5Cforall%20n%5Cin%5Cmathbb%7BZ%7D_%7Bx%3E0%7D%E6%9C%89n%3Dk_nm%2Bl_n%2C0%5Cleq%20l_n%3Cm$

$%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%3D%5CVert%20x%5E%7Bk_nm%2Bl_n%7D%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%5Cleq%5CVert%20x%5CVert%5E%7B%5Cfrac%7Bl_n%7D%7Bn%7D%7D%5CVert%20x%5Em%5CVert%5E%7B%5Cfrac%7Bk_n%7D%7Bn%7D%7D$

$%5Cleq%5CVert%20x%5CVert%5E%7B%5Cfrac%7Bl_n%7D%7Bn%7D%7D(r%2B%5Cvarepsilon)%5E%7B%5Cfrac%7Bmk_n%7D%7Bn%7D%7D$

$%5Cvarlimsup_%7Bn%0A%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%5Cleq%20r%5Cimplies%20%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%3D%5Cinf%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D$

$%5Cmathbb%7BQ.E.D.%7D$

$%5Cmathscr%7BL%7Demma3%EF%BC%9A%E8%AE%BE%5Cmathscr%7BA%7D%E6%98%AF%E5%85%B7%E6%9C%89%E5%8D%95%E4%BD%8D%E5%85%83e%E7%9A%84Banach%E4%BB%A3%E6%95%B0%EF%BC%8Cx%5Cin%5Cmathscr%7BA%7D$

$%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%3C1%2C%E5%88%991%5Cin%20%5Crho(x)%2C%E4%B8%94(e-x)%5E%7B-1%7D%3D%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20x%5En%2C$

$%E4%B8%94%20%5CVert%20x%5CVert%20%3C1%E6%97%B6%5CVert(e-x)%5E%7B-1%7D%5CVert%5Cleq%5Cfrac%7B1%7D%7B1-%5CVert%20x%5CVert%7D.$

证明:

$%E5%8F%96%20%5Cvarepsilon%3E0%E4%BD%BF%E5%BE%97r%2B%5Cvarepsilon%3C1.%E5%8F%96%E8%87%AA%E7%84%B6%E6%95%B0N%E4%BD%BF%E5%BE%97m%5Cgeq%20n%5Cgeq%20N%E6%9C%89$

$%5CVert%5Csum_%7Bk%3Dn%7D%5Em%20x%5Ek%5CVert%5Cleq%5Csum_%7Bk%3Dn%7D%5Em%20%5CVert%20x%5Ek%5CVert%5Cleq%5Csum_%7Bk%3Dn%7D%5Em%20(r%2B%5Cvarepsilon)%5Ek%5Cto0$

$%E8%AE%B0y_n%3D%5Csum_%7Bk%3D0%7D%5En%20x%5Ek%5C%20%2C%5Clim_%7Bn%5Cto%20%5Cinfty%7Dy_n%3D%5Csum_%7Bk%3D0%7D%5E%5Cinfty%20x%5Ek%3Dy$

$(x-%20e)y_n%3Dy_n(x-%20e)%3De-x%5E%7Bn%2B1%7D$

$%20%5CVert%20x%5En%5CVert%5Cleq%5C%20(r%2B%5Cvarepsilon)%5En%5Cto0$

$(x-%20e)y%3Dy(x-e)%3De$

$1%5Cin%20%5Crho(x)%2C%E4%B8%94(e-x)%5E%7B-1%7D%3D%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20x%5En.$

$%E5%BD%93%20%5CVert%20x%5CVert%20%3C1%E6%97%B6%5CVert(e-x)%5E%7B-1%7D%5CVert%5Cleq%5Csum_%7Bk%3D0%7D%5E%5Cinfty%20%5CVert%20x%5Ek%5CVert%3D%5Cfrac%7B1%7D%7B1-%5CVert%20x%5CVert%7D$

$%5Cmathbb%7BQ.E.D.%7D$

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$%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%3Dr%2C%E5%88%99%5C%7B%5Clambda%20%EF%BC%9A%20%5Cvert%20%5Clambda%20%5Cvert%3Er%5C%7D%5Csubset%20%5Crho(x)$

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$%5Cvert%20%5Clambda%5Cvert%20%3E%5CVert%20x%5CVert%E6%97%B6%EF%BC%8C$

$%5CVert(%5Clambda%20e-x)%5E%7B-1%7D%5CVert%5Cleq%5Cfrac%7B1%7D%7B%5Cvert%20%5Clambda%5Cvert-%5CVert%20x%5CVert%7D.$

证明:

$%E5%AF%B9%E4%BA%8E%20%5Clambda%5Cneq0%2C%5Clambda%20e-%20x%3D%5Clambda%20(e-%20%5Cfrac%7Bx%7D%7B%5Clambda%7D)$

$%5Clambda%20%20%5Cin%20%5Crho(x)%5CRightarrow%201%5Cin%5Crho(%5Cfrac%7Bx%7D%7B%5Clambda%7D)$

$%5Cmathbb%7BQ.E.D.%7D$

$%5Cmathscr%7BD%7Defination5%EF%BC%9A%E8%AE%BE%5Cmathscr%7BA%7D%E6%98%AF%E5%85%B7%E6%9C%89%E5%8D%95%E4%BD%8D%E5%85%83e%E7%9A%84Banach%E4%BB%A3%E6%95%B0$

$x%5Cin%20%5Cmathscr%7BA%7D%2C%E6%88%91%E4%BB%AC%E7%A7%B0r(x)%3D%5Cmax_%7B%5Clambda%5Cin%5Csigma(x)%7D%5Cvert%20%5Clambda%5Cvert%E4%B8%BAx%E7%9A%84%E8%B0%B1%E5%8D%8A%E5%BE%84.$

$%5Cmathscr%7BT%7Deorem6(Gelfand%E5%AE%9A%E7%90%86)%EF%BC%9A%E8%AE%BE%5Cmathscr%7BA%7D%E6%98%AF%E5%85%B7%E6%9C%89%E5%8D%95%E4%BD%8D%E5%85%83e%E7%9A%84Banach%E4%BB%A3%E6%95%B0$

$x%5Cin%20%5Cmathscr%7BA%7D%2C%E5%88%99%E8%B0%B1%E5%8D%8A%E5%BE%84r(x)%3D%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D.$

证明:

$%E5%8D%B3%E8%AF%81r(x)%5Cgeq%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%E5%8D%B3%E5%8F%AF$

$%5Cvert%20%5Clambda%20%5Cvert%3E%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%E6%97%B6%2C%E8%AE%B0a%3Dr(x)$

$(%5Clambda%20e-%20x)%5E%7B-1%7D%3D%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20%5Cfrac%7Bx%5En%7D%7B%5Clambda%5E%7Bn%2B1%7D%7D%2C$

$%5Cforall%20f%5Cin%20%5Cmathscr%7BA%7D%5E*%E6%9C%89$

$f((%5Clambda%20e-%20x)%5E%7B-1%7D)%3D%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20%5Cfrac%7Bf(x%5En)%7D%7B%5Clambda%5E%7Bn%2B1%7D%7D%2C$

$%E5%AE%B9%E6%98%93%E7%9F%A5%E9%81%93%5Cforall%20%20%5Cvarepsilon%3E0%EF%BC%8C$

$%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20%5Cfrac%7B%7Cf(x%5En)%7C%7D%7B(a%2B%5Cvarepsilon)%5E%7Bn%2B1%7D%7D%3C%2B%5Cinfty$

$%5Csup_n%20%5CBig%7Cf%5CBig(%5Cfrac%7Bx%5En%7D%7B(a%2B%5Cvarepsilon)%5E%7Bn%2B1%7D%7D%5CBig)%5CBig%7C%3C%2B%5Cinfty$

由共鸣定理得

$M%3D%5Csup_n%20%5CBig%5CVert%5Cfrac%7Bx%5En%7D%7B(a%2B%5Cvarepsilon)%5E%7Bn%2B1%7D%7D%5CBig%5CVert%3C%2B%5Cinfty$

$%5CVert%20x%5En%5CVert%3CM(a%2B%5Cvarepsilon)%5En$

$%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D%5Cleq%20a%2B%5Cvarepsilon$

$%E6%98%BE%E7%84%B6r(x)%5Cgeq%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D$

$%E6%89%80%E4%BB%A5%E8%B0%B1%E5%8D%8A%E5%BE%84r(x)%3D%5Cmax_%7B%5Clambda%5Cin%5Csigma(x)%7D%5Cvert%20%5Clambda%5Cvert%3D%5Clim_%7Bn%5Cto%20%5Cinfty%7D%5CVert%20x%5En%5CVert%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D.$

$%5Cmathbb%7BQ.E.D.%7D$

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