Prove $Tx=x$, for $xin H$, if and only if $(Tx,x)=|x|^2$ and $ker(I-T)=ker(I-T^*)$












0












$begingroup$



Let $H$ be a complex Hilbert space and $T:Hrightarrow H$ an operation such that $|T|leq 1$. Show that





  1. $Tx=x$ if and only if $(Tx,x)=|x|^2$


  2. $ker(I-T)=ker(I-T^*)$.




My attempt

1. $(Tx,x)=(x,x)=|x|^2$ if $Tx=x$. Conversely, WTS $|Tx-x|=0$ $forall xin H$.



We get $|Tx-x|=(Tx-x,Tx-x)=|Tx|^2-(x,Tx)$. But $|T|leq 1$ has not been used.



Please how do I proceed?






linear-algebra functional-analysis operator-theory hilbert-spaces adjoint-operators







share|cite|improve this question











$endgroup$

















    0












    $begingroup$



    Let $H$ be a complex Hilbert space and $T:Hrightarrow H$ an operation such that $|T|leq 1$. Show that





    1. $Tx=x$ if and only if $(Tx,x)=|x|^2$


    2. $ker(I-T)=ker(I-T^*)$.




    My attempt

    1. $(Tx,x)=(x,x)=|x|^2$ if $Tx=x$. Conversely, WTS $|Tx-x|=0$ $forall xin H$.



    We get $|Tx-x|=(Tx-x,Tx-x)=|Tx|^2-(x,Tx)$. But $|T|leq 1$ has not been used.



    Please how do I proceed?






    linear-algebra functional-analysis operator-theory hilbert-spaces adjoint-operators







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$



      Let $H$ be a complex Hilbert space and $T:Hrightarrow H$ an operation such that $|T|leq 1$. Show that





      1. $Tx=x$ if and only if $(Tx,x)=|x|^2$


      2. $ker(I-T)=ker(I-T^*)$.




      My attempt

      1. $(Tx,x)=(x,x)=|x|^2$ if $Tx=x$. Conversely, WTS $|Tx-x|=0$ $forall xin H$.



      We get $|Tx-x|=(Tx-x,Tx-x)=|Tx|^2-(x,Tx)$. But $|T|leq 1$ has not been used.



      Please how do I proceed?






      linear-algebra functional-analysis operator-theory hilbert-spaces adjoint-operators







      share|cite|improve this question











      $endgroup$





      Let $H$ be a complex Hilbert space and $T:Hrightarrow H$ an operation such that $|T|leq 1$. Show that





      1. $Tx=x$ if and only if $(Tx,x)=|x|^2$


      2. $ker(I-T)=ker(I-T^*)$.




      My attempt

      1. $(Tx,x)=(x,x)=|x|^2$ if $Tx=x$. Conversely, WTS $|Tx-x|=0$ $forall xin H$.



      We get $|Tx-x|=(Tx-x,Tx-x)=|Tx|^2-(x,Tx)$. But $|T|leq 1$ has not been used.



      Please how do I proceed?






      linear-algebra functional-analysis operator-theory hilbert-spaces adjoint-operators




      linear-algebra functional-analysis operator-theory hilbert-spaces adjoint-operators






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 6 at 16:36







      Muhammad Mubarak

















      asked Jan 6 at 15:17









      Muhammad MubarakMuhammad Mubarak

      9810




      9810






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          For the first point, first off since $left langle Tx,xrightrangle=|x|^2$ you also have $left langle x,Txrightrangle = overline{|x|^2}=|x|^2$. Thus
          $$0 leqleftlangle Tx-x,Tx-xrightrangle =|Tx|^2+|x|^2-leftlangle Tx, xrightrangle-leftlangle x,Txrightrangle= |Tx|^2-|x|^2 $$
          Since $|T|leq 1$,
          $$0 leq |Tx|^2-|x|^2 leq |x|^2-|x|^2=0 $$
          and hence $Tx=x$.



          For the second point, using the first point
          begin{align*}xin ker(I-T)iff Tx=x iff left langle Tx,xrightrangle =|x|^2end{align*}



          Now observe that if $|T|leq 1$ then also $|T^*|leq 1$. Indeed,
          begin{align*}|T^*|&=sup_{xneq 0}frac{|T^*x|}{|x|}=sup_{xneq 0}frac{leftlangle T^*x,T^*xrightrangle^{1/2}}{|x|}=sup_{xneq 0}frac{leftlangle TT^*x,xrightrangle^{1/2}}{|x|}leq sup_{xneq 0}frac{|TT^*x|^{1/2}}{|x|^{1/2}}leq\
          &leqsup_{xneq 0}frac{|T^*x|^{1/2}}{|x|^{1/2}}=|T^*|^{1/2}end{align*}
          i.e. $|T^*|leq |T^*|^{1/2}$ and thus $|T^*|leq 1$ (remark: this argument can be easily extended to prove that in general $|T|=|T^*|$ for any bounded linear operator $T$).



          Therefore the first point may be applied to $T^*$, too.



          begin{align*}
          xin ker (I-T^*)iff T^*x=xiff left langle T^*x,xrightrangle = |x|^2iff left langle x,Txrightrangle =|x|^2
          end{align*}
          But $left langle Tx,xrightrangle =|x|^2 $ implies $left langle x, Txrightrangle = left langle Tx,xrightrangle =|x|^2$, as shown above, and viceversa, thus the equivalence is proved.






          share|cite|improve this answer











          $endgroup$














            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',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            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%2f3063982%2fprove-tx-x-for-x-in-h-if-and-only-if-tx-x-x-2-and-keri-t-ker%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2












            $begingroup$

            For the first point, first off since $left langle Tx,xrightrangle=|x|^2$ you also have $left langle x,Txrightrangle = overline{|x|^2}=|x|^2$. Thus
            $$0 leqleftlangle Tx-x,Tx-xrightrangle =|Tx|^2+|x|^2-leftlangle Tx, xrightrangle-leftlangle x,Txrightrangle= |Tx|^2-|x|^2 $$
            Since $|T|leq 1$,
            $$0 leq |Tx|^2-|x|^2 leq |x|^2-|x|^2=0 $$
            and hence $Tx=x$.



            For the second point, using the first point
            begin{align*}xin ker(I-T)iff Tx=x iff left langle Tx,xrightrangle =|x|^2end{align*}



            Now observe that if $|T|leq 1$ then also $|T^*|leq 1$. Indeed,
            begin{align*}|T^*|&=sup_{xneq 0}frac{|T^*x|}{|x|}=sup_{xneq 0}frac{leftlangle T^*x,T^*xrightrangle^{1/2}}{|x|}=sup_{xneq 0}frac{leftlangle TT^*x,xrightrangle^{1/2}}{|x|}leq sup_{xneq 0}frac{|TT^*x|^{1/2}}{|x|^{1/2}}leq\
            &leqsup_{xneq 0}frac{|T^*x|^{1/2}}{|x|^{1/2}}=|T^*|^{1/2}end{align*}
            i.e. $|T^*|leq |T^*|^{1/2}$ and thus $|T^*|leq 1$ (remark: this argument can be easily extended to prove that in general $|T|=|T^*|$ for any bounded linear operator $T$).



            Therefore the first point may be applied to $T^*$, too.



            begin{align*}
            xin ker (I-T^*)iff T^*x=xiff left langle T^*x,xrightrangle = |x|^2iff left langle x,Txrightrangle =|x|^2
            end{align*}
            But $left langle Tx,xrightrangle =|x|^2 $ implies $left langle x, Txrightrangle = left langle Tx,xrightrangle =|x|^2$, as shown above, and viceversa, thus the equivalence is proved.






            share|cite|improve this answer











            $endgroup$


















              2












              $begingroup$

              For the first point, first off since $left langle Tx,xrightrangle=|x|^2$ you also have $left langle x,Txrightrangle = overline{|x|^2}=|x|^2$. Thus
              $$0 leqleftlangle Tx-x,Tx-xrightrangle =|Tx|^2+|x|^2-leftlangle Tx, xrightrangle-leftlangle x,Txrightrangle= |Tx|^2-|x|^2 $$
              Since $|T|leq 1$,
              $$0 leq |Tx|^2-|x|^2 leq |x|^2-|x|^2=0 $$
              and hence $Tx=x$.



              For the second point, using the first point
              begin{align*}xin ker(I-T)iff Tx=x iff left langle Tx,xrightrangle =|x|^2end{align*}



              Now observe that if $|T|leq 1$ then also $|T^*|leq 1$. Indeed,
              begin{align*}|T^*|&=sup_{xneq 0}frac{|T^*x|}{|x|}=sup_{xneq 0}frac{leftlangle T^*x,T^*xrightrangle^{1/2}}{|x|}=sup_{xneq 0}frac{leftlangle TT^*x,xrightrangle^{1/2}}{|x|}leq sup_{xneq 0}frac{|TT^*x|^{1/2}}{|x|^{1/2}}leq\
              &leqsup_{xneq 0}frac{|T^*x|^{1/2}}{|x|^{1/2}}=|T^*|^{1/2}end{align*}
              i.e. $|T^*|leq |T^*|^{1/2}$ and thus $|T^*|leq 1$ (remark: this argument can be easily extended to prove that in general $|T|=|T^*|$ for any bounded linear operator $T$).



              Therefore the first point may be applied to $T^*$, too.



              begin{align*}
              xin ker (I-T^*)iff T^*x=xiff left langle T^*x,xrightrangle = |x|^2iff left langle x,Txrightrangle =|x|^2
              end{align*}
              But $left langle Tx,xrightrangle =|x|^2 $ implies $left langle x, Txrightrangle = left langle Tx,xrightrangle =|x|^2$, as shown above, and viceversa, thus the equivalence is proved.






              share|cite|improve this answer











              $endgroup$
















                2












                2








                2





                $begingroup$

                For the first point, first off since $left langle Tx,xrightrangle=|x|^2$ you also have $left langle x,Txrightrangle = overline{|x|^2}=|x|^2$. Thus
                $$0 leqleftlangle Tx-x,Tx-xrightrangle =|Tx|^2+|x|^2-leftlangle Tx, xrightrangle-leftlangle x,Txrightrangle= |Tx|^2-|x|^2 $$
                Since $|T|leq 1$,
                $$0 leq |Tx|^2-|x|^2 leq |x|^2-|x|^2=0 $$
                and hence $Tx=x$.



                For the second point, using the first point
                begin{align*}xin ker(I-T)iff Tx=x iff left langle Tx,xrightrangle =|x|^2end{align*}



                Now observe that if $|T|leq 1$ then also $|T^*|leq 1$. Indeed,
                begin{align*}|T^*|&=sup_{xneq 0}frac{|T^*x|}{|x|}=sup_{xneq 0}frac{leftlangle T^*x,T^*xrightrangle^{1/2}}{|x|}=sup_{xneq 0}frac{leftlangle TT^*x,xrightrangle^{1/2}}{|x|}leq sup_{xneq 0}frac{|TT^*x|^{1/2}}{|x|^{1/2}}leq\
                &leqsup_{xneq 0}frac{|T^*x|^{1/2}}{|x|^{1/2}}=|T^*|^{1/2}end{align*}
                i.e. $|T^*|leq |T^*|^{1/2}$ and thus $|T^*|leq 1$ (remark: this argument can be easily extended to prove that in general $|T|=|T^*|$ for any bounded linear operator $T$).



                Therefore the first point may be applied to $T^*$, too.



                begin{align*}
                xin ker (I-T^*)iff T^*x=xiff left langle T^*x,xrightrangle = |x|^2iff left langle x,Txrightrangle =|x|^2
                end{align*}
                But $left langle Tx,xrightrangle =|x|^2 $ implies $left langle x, Txrightrangle = left langle Tx,xrightrangle =|x|^2$, as shown above, and viceversa, thus the equivalence is proved.






                share|cite|improve this answer











                $endgroup$



                For the first point, first off since $left langle Tx,xrightrangle=|x|^2$ you also have $left langle x,Txrightrangle = overline{|x|^2}=|x|^2$. Thus
                $$0 leqleftlangle Tx-x,Tx-xrightrangle =|Tx|^2+|x|^2-leftlangle Tx, xrightrangle-leftlangle x,Txrightrangle= |Tx|^2-|x|^2 $$
                Since $|T|leq 1$,
                $$0 leq |Tx|^2-|x|^2 leq |x|^2-|x|^2=0 $$
                and hence $Tx=x$.



                For the second point, using the first point
                begin{align*}xin ker(I-T)iff Tx=x iff left langle Tx,xrightrangle =|x|^2end{align*}



                Now observe that if $|T|leq 1$ then also $|T^*|leq 1$. Indeed,
                begin{align*}|T^*|&=sup_{xneq 0}frac{|T^*x|}{|x|}=sup_{xneq 0}frac{leftlangle T^*x,T^*xrightrangle^{1/2}}{|x|}=sup_{xneq 0}frac{leftlangle TT^*x,xrightrangle^{1/2}}{|x|}leq sup_{xneq 0}frac{|TT^*x|^{1/2}}{|x|^{1/2}}leq\
                &leqsup_{xneq 0}frac{|T^*x|^{1/2}}{|x|^{1/2}}=|T^*|^{1/2}end{align*}
                i.e. $|T^*|leq |T^*|^{1/2}$ and thus $|T^*|leq 1$ (remark: this argument can be easily extended to prove that in general $|T|=|T^*|$ for any bounded linear operator $T$).



                Therefore the first point may be applied to $T^*$, too.



                begin{align*}
                xin ker (I-T^*)iff T^*x=xiff left langle T^*x,xrightrangle = |x|^2iff left langle x,Txrightrangle =|x|^2
                end{align*}
                But $left langle Tx,xrightrangle =|x|^2 $ implies $left langle x, Txrightrangle = left langle Tx,xrightrangle =|x|^2$, as shown above, and viceversa, thus the equivalence is proved.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Jan 6 at 16:18

























                answered Jan 6 at 15:49









                Lorenzo QuarisaLorenzo Quarisa

                3,765623




                3,765623






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3063982%2fprove-tx-x-for-x-in-h-if-and-only-if-tx-x-x-2-and-keri-t-ker%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    -

                    Popular posts from this blog

                    Wiesbaden

                    Image
                    Wappen Deutschlandkarte 50.0825 8.24 117 Koordinaten: 50° 5′  N , 8° 14′  O Basisdaten Bundesland: Hessen Regierungsbezirk: Darmstadt Höhe: 117 m ü. NHN Fläche: 203,93 km 2 Einwohner: 278.654 (31. Dez. 2017) [1] Bevölkerungsdichte: 1366 Einwohner je km 2 Postleitzahlen: 65183–65207, 55246, 55252 Vorlage:Infobox Gemeinde in Deutschland/Wartung/PLZ enthält Text Vorwahlen: 0611, 06122, 06127, 06134 Kfz-Kennzeichen: WI Gemeindeschlüssel: 06 4 14 000 LOCODE: DE WIB NUTS: DE714 Stadtgliederung: 26 Ortsbezirke Adresse der Stadtverwaltung: Schlossplatz 6 65183 Wiesbaden Webpräsenz: www.wiesbaden.de Oberbürgermeister: Sven Gerich (SPD) Lage der Landeshauptstadt Wiesbaden in Hessen und im Regierungsbezirk Darmstadt Offizielles Logo der Landeshauptstadt Wiesbaden Flagge der Landeshauptstadt Wiesbaden Wiesbaden is
                    Read more

                    Marschland

                    Image
                    Kilometerweite Marsch in den Vier- und Marschlanden am Elbdeich Die Wedeler Marschlandschaft in Schleswig-Holstein Der Hadelner Kanal zur Entwässerung der Marsch Die tief gelegene Marsch und ein Entwässerungskanal Das Schöpfwerk Otterndorf im Land Hadeln zur Entwässerung der Medem verfügte damals bereits über die größte Kreiselpumpe Europas. Das Schöpfwerk in Neuhaus für die Aue und des Neuhaus-Bülkauer Kanals Als Marsch(land) (v. niederdt., altsächs. mersc ) – auch Masch , Mersch oder Schwemmland genannt – bezeichnet man eine nacheiszeitlich entstandene geomorphologische Landform im Gebiet der nordwestdeutschen Küsten und Flüsse sowie vergleichbare Landformen weltweit. Inhaltsverzeichnis 1 Landschaft 2 Entstehung 3 Marschböden 4 Nutzung 5 Marschgebiete in Deutschland 6 Siehe auch 7 Literatur 8 Weblinks 9 Einzelnachweise Landschaft | Marschen sind generell flache Landstriche ohne natürliche Erhebungen. Si
                    Read more

                    Dieringhausen

                    Image
                    Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
                    Read more