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What are the approximate eigenvalues of the right shift operator $R$ on $ell_infty$



The Next CEO of Stack OverflowApproximate eigenvalue and continuous spectrumWhat are the Eigenvectors of the curl operator?Spectral theory - continuous spectrumEigenvalues of Left Shift + Right Shift in $l_2([0,infty))$Eigenvalues of the circle over the Laplacian operatorIf an operator has a discrete spectrum eigenvalues, but infinitely many eigenvalues, are these necessarily countable?Spectrum of the right-shift operator on $ell ^2 (mathbbC)$, and a general spectrum questionAre the poles of the resolvent of a Hermitian operator real?The spectrum of a matrix operator is the set of its eigenvalues. But how to prove?Intuitive meaning of the members of the spectrum of a linear operator which are not eigenvalues










3












$begingroup$


I have shown that the spectrum of $R=z$.



Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $forall |z|=1$, $z$ is an approx. eigenvalue. However, are these the only ones?



If they are not how do I find the rest?



Thanks.










share|cite|improve this question











$endgroup$
















    3












    $begingroup$


    I have shown that the spectrum of $R=z$.



    Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $forall |z|=1$, $z$ is an approx. eigenvalue. However, are these the only ones?



    If they are not how do I find the rest?



    Thanks.










    share|cite|improve this question











    $endgroup$














      3












      3








      3





      $begingroup$


      I have shown that the spectrum of $R=z$.



      Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $forall |z|=1$, $z$ is an approx. eigenvalue. However, are these the only ones?



      If they are not how do I find the rest?



      Thanks.










      share|cite|improve this question











      $endgroup$




      I have shown that the spectrum of $R=z$.



      Also, elements on the boundary of the spectrum are approximate eigenvalues, i.e. $forall |z|=1$, $z$ is an approx. eigenvalue. However, are these the only ones?



      If they are not how do I find the rest?



      Thanks.







      functional-analysis operator-theory spectral-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 18 at 11:59









      Andrews

      1,2812422




      1,2812422










      asked Mar 18 at 11:09









      Jhon DoeJhon Doe

      683414




      683414




















          1 Answer
          1






          active

          oldest

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          2












          $begingroup$

          For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have



          $$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$



          Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.



          From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$






          share|cite|improve this answer









          $endgroup$













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            1 Answer
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            active

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            2












            $begingroup$

            For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have



            $$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$



            Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.



            From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$






            share|cite|improve this answer









            $endgroup$

















              2












              $begingroup$

              For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have



              $$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$



              Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.



              From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$






              share|cite|improve this answer









              $endgroup$















                2












                2








                2





                $begingroup$

                For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have



                $$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$



                Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.



                From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$






                share|cite|improve this answer









                $endgroup$



                For all $x in ell_infty$ we have $||Rx||=||x||.$ Thus, for $lambda$ with $|lambda | le 1$ we have



                $$(*) ||Rx-lambda x|| ge | quad ||Rx||-|lambda| ||x|| quad|=(1-|lambda|)||x||.$$



                Now suppose that $ lambda $ is an approximate eigenvalue . Then there is a sequence $(x_n)$ in $ell_infty$ such that $||x_n||=1$ for all $n$ and $(R-lambda I )x_n to 0$.



                From $(*)$ we get: $(1-|lambda|)||x_n|| to 0.$ Since $||x_n||=1$ for all $n$, we derive $|lambda|=1.$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Mar 18 at 12:18









                FredFred

                48.8k11849




                48.8k11849



























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