Existence of inverses of linear combinations of bounded operators

Multi tool use
Multi tool use

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite
2












If $V$ is a banach space and $T$ is a bounded linear operator on $V$. If $|T|< a$ where $a>0$, then is $A=T-aI$ invertible? If so why? We know that if $|T|<1$ then $(I-T)$ is invertible.






functional-analysis operator-theory linear-transformations







share|cite|improve this question



























    up vote
    1
    down vote

    favorite
    2












    If $V$ is a banach space and $T$ is a bounded linear operator on $V$. If $|T|< a$ where $a>0$, then is $A=T-aI$ invertible? If so why? We know that if $|T|<1$ then $(I-T)$ is invertible.






    functional-analysis operator-theory linear-transformations







    share|cite|improve this question

























      up vote
      1
      down vote

      favorite
      2









      up vote
      1
      down vote

      favorite
      2






      2





      If $V$ is a banach space and $T$ is a bounded linear operator on $V$. If $|T|< a$ where $a>0$, then is $A=T-aI$ invertible? If so why? We know that if $|T|<1$ then $(I-T)$ is invertible.






      functional-analysis operator-theory linear-transformations







      share|cite|improve this question















      If $V$ is a banach space and $T$ is a bounded linear operator on $V$. If $|T|< a$ where $a>0$, then is $A=T-aI$ invertible? If so why? We know that if $|T|<1$ then $(I-T)$ is invertible.






      functional-analysis operator-theory linear-transformations




      functional-analysis operator-theory linear-transformations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 14 mins ago









      qbert

      21.2k32356




      21.2k32356










      asked 24 mins ago









      user593295

      335




      335




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          3
          down vote













          Note
          $$
          T-aI=-a(I-frac1aT)
          $$
          and by assumption
          $$
          left|left| frac1aT right|right|=frac1a||T||<1
          $$
          So $(I-frac1aT)^-1$ is invertible and the inverse of $T-aI$ is then
          $$
          -frac1a(I-frac1aT)^-1
          $$






          share|cite|improve this answer




















          • I assume if $(1-1/aT)$ is invertible, then any scale of that operator is invertible? (i.e we can freely multiply by $-a$ and its fine)
            – user593295
            12 mins ago










          • yes, and you can explicitly give its inverse, as in the above
            – qbert
            3 mins ago











          Your Answer




          StackExchange.ifUsing("editor", function ()
          return StackExchange.using("mathjaxEditing", function ()
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          );
          );
          , "mathjax-editing");

          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "69"
          ;
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function()
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled)
          StackExchange.using("snippets", function()
          createEditor();
          );

          else
          createEditor();

          );

          function createEditor()
          StackExchange.prepareEditor(
          heartbeatType: 'answer',
          convertImagesToLinks: true,
          noModals: false,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          );



          );













           

          draft saved


          draft discarded


















          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2975756%2fexistence-of-inverses-of-linear-combinations-of-bounded-operators%23new-answer', 'question_page');

          );

          Post as a guest

















          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          3
          down vote













          Note
          $$
          T-aI=-a(I-frac1aT)
          $$
          and by assumption
          $$
          left|left| frac1aT right|right|=frac1a||T||<1
          $$
          So $(I-frac1aT)^-1$ is invertible and the inverse of $T-aI$ is then
          $$
          -frac1a(I-frac1aT)^-1
          $$






          share|cite|improve this answer




















          • I assume if $(1-1/aT)$ is invertible, then any scale of that operator is invertible? (i.e we can freely multiply by $-a$ and its fine)
            – user593295
            12 mins ago










          • yes, and you can explicitly give its inverse, as in the above
            – qbert
            3 mins ago















          up vote
          3
          down vote













          Note
          $$
          T-aI=-a(I-frac1aT)
          $$
          and by assumption
          $$
          left|left| frac1aT right|right|=frac1a||T||<1
          $$
          So $(I-frac1aT)^-1$ is invertible and the inverse of $T-aI$ is then
          $$
          -frac1a(I-frac1aT)^-1
          $$






          share|cite|improve this answer




















          • I assume if $(1-1/aT)$ is invertible, then any scale of that operator is invertible? (i.e we can freely multiply by $-a$ and its fine)
            – user593295
            12 mins ago










          • yes, and you can explicitly give its inverse, as in the above
            – qbert
            3 mins ago













          up vote
          3
          down vote










          up vote
          3
          down vote









          Note
          $$
          T-aI=-a(I-frac1aT)
          $$
          and by assumption
          $$
          left|left| frac1aT right|right|=frac1a||T||<1
          $$
          So $(I-frac1aT)^-1$ is invertible and the inverse of $T-aI$ is then
          $$
          -frac1a(I-frac1aT)^-1
          $$






          share|cite|improve this answer












          Note
          $$
          T-aI=-a(I-frac1aT)
          $$
          and by assumption
          $$
          left|left| frac1aT right|right|=frac1a||T||<1
          $$
          So $(I-frac1aT)^-1$ is invertible and the inverse of $T-aI$ is then
          $$
          -frac1a(I-frac1aT)^-1
          $$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 18 mins ago









          qbert

          21.2k32356




          21.2k32356











          • I assume if $(1-1/aT)$ is invertible, then any scale of that operator is invertible? (i.e we can freely multiply by $-a$ and its fine)
            – user593295
            12 mins ago










          • yes, and you can explicitly give its inverse, as in the above
            – qbert
            3 mins ago

















          • I assume if $(1-1/aT)$ is invertible, then any scale of that operator is invertible? (i.e we can freely multiply by $-a$ and its fine)
            – user593295
            12 mins ago










          • yes, and you can explicitly give its inverse, as in the above
            – qbert
            3 mins ago
















          I assume if $(1-1/aT)$ is invertible, then any scale of that operator is invertible? (i.e we can freely multiply by $-a$ and its fine)
          – user593295
          12 mins ago




          I assume if $(1-1/aT)$ is invertible, then any scale of that operator is invertible? (i.e we can freely multiply by $-a$ and its fine)
          – user593295
          12 mins ago












          yes, and you can explicitly give its inverse, as in the above
          – qbert
          3 mins ago





          yes, and you can explicitly give its inverse, as in the above
          – qbert
          3 mins ago


















           

          draft saved


          draft discarded















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2975756%2fexistence-of-inverses-of-linear-combinations-of-bounded-operators%23new-answer', 'question_page');

          );

          Post as a guest
































































          aBOQ5wOGZ4Ek9dL2s69tnekq9J7ib81TfUIOhT
          -
          2rM4 x,Dm,aWDeorXo EuvZUVXzOOd,celTNO,vmrqMVmxYuh9A5SbkP9,T

          Popular posts from this blog

          How to check contact read email or not when send email to Individual?

          Image
          Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite I'm using WordPress 4.9.8, CiviCRM to 5.5.1, I usually send email to contact by Search> Find contacts View contact details Action> Send email Send email ok, Contact received mail ok like picture But status only Email sent though contact read email or not. So, can CiviCRM can change status to Email read when contact read email? wordpress email share | improve this question asked Sep 26 at 0:12 ToanLuong 49 9 add a comment  |  up vote 1 down vote favorite I'm using WordPress 4.9.8, CiviCRM to 5.5.1, I usually send email to contact by Search> Find contacts View contact details Action> Send email Send email ok, Contact received mail ok like picture But status only Email sent though contact read email or not. So, can CiviCRM can change status to Email read when contact read email? wordpress email share | improve this questi...
          Read more

          How many registers does an x86_64 CPU actually have?

          Image
          Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite I am currently learning reverse engineering and am studying the flags register. I had in my mind that rflags was just another name for one of the 16 general purpose registers, for example rax or rbx . But it looks like rflags is actually an additional register. So that makes 17 registers in total... how many more could there be? I have spent at least an hour on this and found numerous different answers. The best answer so far is this, which says that there are 40 registers in total. 16 General Purpose Registers 2 Status Registers 6 Code Segment Registers 16 SSE Registers 8 FPU/MMX Registers But if I add that up, I get 48. Could anybody provide an official answer on how many registers an x86_64 CPU has (e.g. an Intel i7). Additionally, I have seen references to 'hardware' and 'architectural' registers. What are those registers and how many are there? register x86-64 share | improve this...
          Read more

          Displaying single band from multi-band raster using QGIS

          Image
          Clash Royale CLAN TAG #URR8PPP 1 How can I extract a single band from multi-band raster in QGIS? I have an remote sensed image which has 6 bands (including NDVI band), I want to display each band separately, but have no idea how to do. I have seen some questions similar here but none worked for me. The original image (has 6 bands) is: I want to display the band 6 which should be like this: But I tried gdal_translate, and couldn't get the correct result. What I have got is: qgis raster multi-band share | improve this question edited Mar 5 at 0:53 Summer asked Mar 4 at 6:42 Summer Summer 23 6 Is this any help gis.stackexchange.com/questions/220658/… ? if not gis.stackexchange.com/questions/62133/… might help. – Michael Stimson Mar 4 at 6:46 Thanks for answering but when I used gdal_translate, qgis showed that 'Error 4: Kayena.tif: No such file or directory". Would you know how to fi...
          Read more