Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

image/svg+xml

Contents

Frontmatter

I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

IV. The Spectral Theorem

We begin with two elementary matters that could have been discussed much earlier, namely: the partial ordering of self-adjoint operators in bounded linear operators(HHilbert space) \( \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \); and projections, i.e. self-adjoint idempotents. Filling in the details below is recommended as an exercise.

Write bounded linear operators(HHilbert space) SAselfadjoint elements=equals{setAself-adjoint operatorelement ofbounded linear operators(HHilbert space)|such thatAself-adjoint operator=equalsAself-adjoint operator*}set \( { \mathcal{L}\mathopen{}\left( H\right)\mathclose{} }_{\rm SA}= \mathopen{}\left\{\, A\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}\,\middle\vert\, A= A^{*}\,\right\}\mathclose{} \). For Aself-adjoint operator\( A \) and Bself-adjoint operator\( B \) in bounded linear operators(HHilbert space) SAselfadjoint elements \( { \mathcal{L}\mathopen{}\left( H\right)\mathclose{} }_{\rm SA} \), say Aself-adjoint operatorless than or equal toBself-adjoint operator \( A\leq B \) provided Bself-adjoint operator-minusAself-adjoint operatorgreater than or equal to0zero \( B-A\geq 0 \), i.e. Aself-adjoint operator(xvector), xvectorless than or equal toBself-adjoint operator(xvector), xvector \( \mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}\leq \mathopen{}\left\langle{}B\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{} \) for all xvectorelement ofHHilbert space \( x\in H \). Notice for Toperatorelement ofbounded linear operators(HHilbert space) \( T\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \), Aself-adjoint operatorless than or equal toBself-adjoint operator \( A\leq B \) implies Toperator*timesAself-adjoint operatortimesToperatorless than or equal toToperator*timesBself-adjoint operatortimesToperator \( T^{*}AT\leq T^{*}BT \) because Toperator*(Aself-adjoint operator(Toperator(xvector))), xvector=equalsAself-adjoint operator(Toperator(xvector)), Toperator(xvector)less than or equal toBself-adjoint operator(Toperator(xvector)), xvector \( \mathopen{}\left\langle{} T^{*}\mathopen{}\left( A\mathopen{}\left( T\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}, x\right\rangle\mathclose{}= \mathopen{}\left\langle{}A\mathopen{}\left( T\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, T\mathopen{}\left( x\right)\mathclose{}\right\rangle\mathclose{}\leq \mathopen{}\left\langle{}B\mathopen{}\left( T\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, x\right\rangle\mathclose{} \) for all xvectorelement ofHHilbert space \( x\in H \). For Aself-adjoint operatorelement of bounded linear operators(HHilbert space) SAselfadjoint elements \( A\in { \mathcal{L}\mathopen{}\left( H\right)\mathclose{} }_{\rm SA} \), recall from functional calculus fcontinuous functionconverges tofcontinuous function(Aself-adjoint operator) \( f \to f\mathopen{}\left( A\right)\mathclose{} \) from Cspace of continuous functions(σ(Aself-adjoint operator)) \( \mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \) to bounded linear operators(HHilbert space) \( \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \), fcontinuous functiongreater than or equal to0zero \( f\geq 0 \) on σ(Aself-adjoint operator) \( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{} \) implies fcontinuous function(Aself-adjoint operator)greater than or equal to0zero \( f\mathopen{}\left( A\right)\mathclose{}\geq 0 \) because fcontinuous function(Aself-adjoint operator)=equals (fcontinuous function1one2two(Aself-adjoint operator)) 2two =equals (fcontinuous function1one2two(Aself-adjoint operator)) *timesfcontinuous function1one2two(Aself-adjoint operator) \( f\mathopen{}\left( A\right)\mathclose{}= {\mathopen{}\left({f}^{\frac{1}{2}}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}}^{2}= \mathopen{}\left({f}^{\frac{1}{2}}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} ^{*}{f}^{\frac{1}{2}}\mathopen{}\left( A\right)\mathclose{} \) so fcontinuous function\( f \) and gcontinuous function\( g \) in Cspace of continuous functions(σ(Aself-adjoint operator)Rreal numbers) \( \mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}, \mathbb{R}\right)\mathclose{} \) with fcontinuous functionless than or equal togcontinuous function \( f\leq g \) implies fcontinuous function(Aself-adjoint operator)less than or equal togcontinuous function(Aself-adjoint operator) \( f\mathopen{}\left( A\right)\mathclose{}\leq g\mathopen{}\left( A\right)\mathclose{} \).

Definition IV.1

Pprojectionelement ofbounded linear operators(HHilbert space) \( P\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) is a projection if Pprojection=equalsPprojection*=equalsPprojection2two \( P= P^{*}= {P}^{2} \).

For a closed subspace Eclosed subspacesubsetHHilbert space \( E\subseteq H \), the orthogonal projection of HHilbert space\( H \) onto Eclosed subspace\( E \) is a projection in this sense. If Pprojection\( P \) is a projection, then Pprojection(HHilbert space) \( P\mathopen{}\left( H\right)\mathclose{} \) is closed and Pprojection:mapsHHilbert spacetoPprojection(HHilbert space) \( P : H \to P\mathopen{}\left( H\right)\mathclose{} \) is the orthogonal projection of HHilbert space\( H \) on Pprojection(HHilbert space) \( P\mathopen{}\left( H\right)\mathclose{} \). For projections Pprojection\( P \) and Qprojection\( Q \) we have Pprojectionless than or equal toQprojection \( P\leq Q \) if and only if Pprojection(HHilbert space)subsetQprojection(HHilbert space) \( P\mathopen{}\left( H\right)\mathclose{}\subseteq Q\mathopen{}\left( H\right)\mathclose{} \) if and only if QprojectiontimesPprojection=equalsPprojection \( QP= P \). In this situation, Qprojection-minusPprojection \( Q-P \) is projection on Qprojection(HHilbert space)intersection (Pprojection(HHilbert space)) \( Q\mathopen{}\left( H\right)\mathclose{}\cap { \mathopen{}\left(P\mathopen{}\left( H\right)\mathclose{}\right)\mathclose{} }^{\perp} \) (i.e., Qprojection(HHilbert space)intersection(1one-minusPprojection)(HHilbert space) \( Q\mathopen{}\left( H\right)\mathclose{}\cap \mathopen{}\left(1-P\right)\mathclose{}\mathopen{}\left( H\right)\mathclose{} \)), often written Qprojection(HHilbert space)Pprojection(HHilbert space) \( Q\mathopen{}\left( H\right)\mathclose{}\ominus P\mathopen{}\left( H\right)\mathclose{} \).

Fix Aself-adjoint operatorelement of bounded linear operators(HHilbert space) SAselfadjoint elements \( A\in { \mathcal{L}\mathopen{}\left( H\right)\mathclose{} }_{\rm SA} \). For λreal numberelement ofRreal numbers \( λ\in \mathbb{R} \), define fcontinuous functionλreal number \( {f}_{λ} \) on Rreal numbers\( \mathbb{R} \) by fcontinuous functionλreal number(treal number)=equalstreal number-minusλreal number+plus|modulustreal number-minusλreal number|modulus \( {f}_{λ}\mathopen{}\left( t\right)\mathclose{}= t-λ+\mathopen{}\left\lvert{}t-λ\right\rvert\mathclose{} \). Regard fcontinuous functionλreal numberelement ofCspace of continuous functions(σ(Aself-adjoint operator)) \( {f}_{λ}\in \mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \). Let Pprojectionλreal number \( {P}_{λ} \) be projection on Kerkernel(fcontinuous functionλreal number(Aself-adjoint operator)) \( \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \).

Lemma IV.2

  1. λreal numberless than or equal toμreal number \( λ\leq μ \) implies Pprojectionλreal numberless than or equal toPprojectionμreal number \( {P}_{λ}\leq {P}_{μ} \).
  2. λreal number<less thanminminimumσ(Aself-adjoint operator) \( λ\lt \min{}\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{} \) implies Pprojectionλreal number=equals0zero \( {P}_{λ}= 0 \).
  3. λreal number>greater thanmaxmaximumσ(Aself-adjoint operator) \( λ\gt \max{}\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{} \) implies Pprojectionλreal number=equals1one \( {P}_{λ}= 1 \).

Proof.

  1. Notice λreal numberless than or equal toμreal number \( λ\leq μ \) implies fcontinuous functionμreal number=equalsgcontinuous functiontimesfcontinuous functionλreal number \( {f}_{μ}= g{f}_{λ} \) for a continuous function gcontinuous function\( g \), so fcontinuous functionμreal number(Aself-adjoint operator)=equalsgcontinuous function(Aself-adjoint operator)timesfcontinuous functionλreal number(Aself-adjoint operator) \( {f}_{μ}\mathopen{}\left( A\right)\mathclose{}= g\mathopen{}\left( A\right)\mathclose{}{f}_{λ}\mathopen{}\left( A\right)\mathclose{} \). Hence fcontinuous functionλreal number(Aself-adjoint operator)(xvector)=equals0zero \( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= 0 \) implies fcontinuous functionμreal number(Aself-adjoint operator)(xvector)=equals0zero \( {f}_{μ}\mathopen{}\left( A\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= 0 \), that is, Kerkernel(fcontinuous functionλreal number(Aself-adjoint operator))subsetKerkernel(fcontinuous functionμreal number(Aself-adjoint operator)) \( \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}\subseteq \operatorname{Ker}\mathopen{}\left( {f}_{μ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \).
  2. λreal number<less thanminminimumσ(Aself-adjoint operator) \( λ\lt \min{}\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{} \) implies Aself-adjoint operator-minusλreal numbergreater than or equal to0zero \( A-λ\geq 0 \) and Aself-adjoint operator-minusλreal number \( A-λ \) invertible, so fcontinuous functionλreal number(Aself-adjoint operator)=equals2twotimes(Aself-adjoint operator-minusλreal number) \( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}= 2\mathopen{}\left(A-λ\right)\mathclose{} \) is invertible, so Pprojectionλreal number=equals0zero \( {P}_{λ}= 0 \).
  3. λreal number>greater thanmaxmaximumσ(Aself-adjoint operator) \( λ\gt \max{}\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{} \), then Aself-adjoint operator-minusλreal numberless than or equal to0zero \( A-λ\leq 0 \) and so |modulusAself-adjoint operator-minusλreal number|modulus=equalsλreal number-minusAself-adjoint operator \( \mathopen{}\left\lvert{}A-λ\right\rvert\mathclose{}= λ-A \) so fcontinuous functionλreal number(Aself-adjoint operator)=equals0zero \( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}= 0 \) so Kerkernel(fcontinuous functionλreal number(Aself-adjoint operator))=equalsHHilbert space \( \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}= H \).

Lemma IV.3

  1. Aself-adjoint operatortimesPprojectionλreal number=equalsPprojectionλreal numbertimesAself-adjoint operator \( A{P}_{λ}= {P}_{λ}A \).
  2. Aself-adjoint operatortimesPprojectionλreal numberless than or equal toλreal numbertimesPprojectionλreal number \( A{P}_{λ}\leq λ{P}_{λ} \).
  3. Aself-adjoint operatortimes(1one-minusPprojectionλreal number)greater than or equal toλreal numbertimes(1one-minusPprojectionλreal number) \( A\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\geq λ\mathopen{}\left(1-{P}_{λ}\right)\mathclose{} \).

Proof.

  1. Let xvectorelement ofKerkernel(fcontinuous functionλreal number(Aself-adjoint operator)) \( x\in \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \). Then fcontinuous functionλreal number(Aself-adjoint operator)(Aself-adjoint operator(xvector))=equalsAself-adjoint operator(fcontinuous functionλreal number(Aself-adjoint operator(xvector)))=equals0zero \( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\mathopen{}\left( A\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}= A\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}= 0 \). So Aself-adjoint operatortimesKerkernel(fcontinuous functionλreal number(Aself-adjoint operator))subsetKerkernel(fcontinuous functionλreal number(Aself-adjoint operator)) \( A\operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}\subseteq \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \), so Pprojectionλreal numbertimesAself-adjoint operatortimesPprojectionλreal number=equalsAself-adjoint operatortimesPprojectionλreal number \( {P}_{λ}A{P}_{λ}= A{P}_{λ} \). But also Pprojectionλreal numbertimesAself-adjoint operator=equalsPprojectionλreal number*timesAself-adjoint operator*=equals(Pprojectionλreal numbertimesAself-adjoint operatortimesPprojectionλreal number)*=equalsPprojectionλreal number*timesAself-adjoint operator*timesPprojectionλreal number*=equalsPprojectionλreal numbertimesAself-adjoint operatortimesPprojectionλreal number=equalsAself-adjoint operatortimesPprojectionλreal number . \[ {P}_{λ}A= {P}_{λ}^{*} A^{*}= \mathopen{}\left({P}_{λ}A{P}_{λ}\right)\mathclose{}^{*}= {P}_{λ}^{*} A^{*} {P}_{λ}^{*}= {P}_{λ}A{P}_{λ}= A{P}_{λ} \text{.} \]
  2. For yvectorelement ofKerkernel(fcontinuous functionλreal number(Aself-adjoint operator)) \( y\in \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \), we have that (Aself-adjoint operator-minusλreal number)timesyvector=equals|modulusAself-adjoint operator-minusλreal number|modulustimesyvector \( \mathopen{}\left(A-λ\right)\mathclose{}y= {-}\mathopen{}\left\lvert{}A-λ\right\rvert\mathclose{}y \). So for any xvectorelement ofHHilbert space \( x\in H \), (Aself-adjoint operator-minusλreal number)(Pprojectionλreal number(xvector)), xvector=equalsPprojectionλreal number((Aself-adjoint operator-minusλreal number)(Pprojectionλreal number(xvector))xvector)=equals(Aself-adjoint operator-minusλreal number)(Pprojectionλreal number(xvector)), Pprojectionλreal number(xvector)=equals |modulusAself-adjoint operator-minusλreal number|modulus(Pprojectionλreal number(xvector)), Pprojectionλreal number(xvector) less than or equal to0zero , \[ \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, x\right\rangle\mathclose{}= \mathopen{}\left\langle{}{P}_{λ}\mathopen{}\left( \mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, x\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right\rangle\mathclose{}= {-} \mathopen{}\left\langle{}\mathopen{}\left\lvert{}A-λ\right\rvert\mathclose{}\mathopen{}\left( {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right\rangle\mathclose{} \leq 0 \text{,} \] showing (Aself-adjoint operator-minusλreal number)(Pprojectionλreal number)less than or equal to0zero \( \mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {P}_{λ}\right)\mathclose{}\leq 0 \).
  3. For any yvectorelement ofHHilbert space \( y\in H \), (Aself-adjoint operator-minusλreal number)(fcontinuous functionλreal number(Aself-adjoint operator1one2two(yvector))), fcontinuous functionλreal number(Aself-adjoint operator1one2two(yvector))=equals(Aself-adjoint operator-minusλreal number)(fcontinuous functionλreal number(Aself-adjoint operator(yvector))), yvector=equals( (Aself-adjoint operator-minusλreal number)2two +plus(Aself-adjoint operator-minusλreal number)times|modulusAself-adjoint operator-minusλreal number|modulus)(yvector), yvectorgreater than or equal to0zero . \[ \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}, {f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}, y\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left({\mathopen{}\left(A-λ\right)\mathclose{}}^{2}+\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left\lvert{}A-λ\right\rvert\mathclose{}\right)\mathclose{}\mathopen{}\left( y\right)\mathclose{}, y\right\rangle\mathclose{}\geq 0 \text{.} \] It follows that (Aself-adjoint operator-minusλreal number)(xvector), xvectorgreater than or equal to0zero \( \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}\geq 0 \) for all xvectorelement of fcontinuous functionλreal number(Aself-adjoint operator1one2two(HHilbert space)) ¯complex conjugate \( x\in \overline{ {f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( H\right)\mathclose{}\right)\mathclose{} } \). Notice fcontinuous functionλreal number(Aself-adjoint operator1one2two(HHilbert space)) ¯complex conjugate=equals (Kerkernel( (fcontinuous functionλreal number(Aself-adjoint operator1one2two)) *)) =equalsKerkernel( (fcontinuous functionλreal number(Aself-adjoint operator1one2two)) )=equals(1one-minusPprojectionλreal number)(HHilbert space) , \[ \overline{ {f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( H\right)\mathclose{}\right)\mathclose{} }= { \mathopen{}\left(\operatorname{Ker}\mathopen{}\left( \mathopen{}\left({f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\right)\mathclose{}\right)\mathclose{} ^{*}\right)\mathclose{}\right)\mathclose{} }^{\perp}= \operatorname{Ker}\mathopen{}\left( { \mathopen{}\left({f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\right)\mathclose{}\right)\mathclose{} }^{\perp}\right)\mathclose{}= \mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left( H\right)\mathclose{} \text{,} \] so 0zeroless than or equal to(Aself-adjoint operator-minusλreal number)((1one-minusPprojectionλreal number)(yvector)), (1one-minusPprojectionλreal number)(yvector)=equals(Aself-adjoint operator-minusλreal number)((1one-minusPprojectionλreal number)(yvector)), yvector . \[ 0\leq \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( \mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}, \mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( \mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}, y\right\rangle\mathclose{} \text{.} \]

Lemma IV.4

For λreal numberless than or equal toμreal number \( λ\leq μ \), we have λreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number)less than or equal toAself-adjoint operatortimes(Pprojectionμreal number-minusPprojectionλreal number)less than or equal toμreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number) \( λ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\leq A\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\leq μ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{} \).

Proof. Continuing with the proof of the spectral theorem (which we will state at the end), fix areal number<less thanbreal number \( a\lt b \) with σ(Aself-adjoint operator)subset(intervalareal number, breal number)interval \( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\subseteq \mathopen{}\left(a, b\right)\mathclose{} \). So (Pprojectionμreal number-minusPprojectionλreal number)timesAself-adjoint operatortimes(1one-minusPprojectionλreal number)times(Pprojectionμreal number-minusPprojectionλreal number)greater than or equal toλreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number)times(1one-minusPprojectionλreal number)times(Pprojectionμreal number-minusPprojectionλreal number) \[ \mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}A\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\geq λ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{} \] implies Aself-adjoint operatortimes(1one-minusPprojectionλreal number)times(Pprojectionμreal number-minusPprojectionλreal number)greater than or equal toλreal numbertimes(1one-minusPprojectionλreal number)times(Pprojectionμreal number-minusPprojectionλreal number)=equalsλreal numbertimes(Pprojectionμreal number-minusPprojectionλreal numbertimesPprojectionμreal number-minusPprojectionλreal number+plusPprojectionλreal number)=equalsλreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number) \[ A\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\geq λ\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}= λ\mathopen{}\left({P}_{μ}-{P}_{λ}{P}_{μ}-{P}_{λ}+{P}_{λ}\right)\mathclose{}= λ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{} \] since Pprojectionλreal numberless than or equal toPprojectionμreal number \( {P}_{λ}\leq {P}_{μ} \). Thus Aself-adjoint operatortimesPprojectionμreal number-minusPprojectionλreal numbergreater than or equal toλreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number) \( A{P}_{μ}-{P}_{λ}\geq λ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{} \). Next, Aself-adjoint operatortimesPprojectionμreal numberless than or equal toμreal numbertimesPprojectionμreal number \( A{P}_{μ}\leq μ{P}_{μ} \), so (Pprojectionμreal number-minusPprojectionλreal number)timesAself-adjoint operatortimesPprojectionμreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number)less than or equal toμreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number)timesPprojectionμreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number) \[ \mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}A{P}_{μ}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\leq μ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}{P}_{μ}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{} \] implies Aself-adjoint operatortimes (Pprojectionμreal number-minusPprojectionλreal number) 2two less than or equal toμreal numbertimes (Pprojectionμreal number-minusPprojectionλreal number) 2two \( A{\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}}^{2}\leq μ{\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}}^{2} \) implies Aself-adjoint operatortimes(Pprojectionμreal number-minusPprojectionλreal number)less than or equal toμreal numbertimes(Pprojectionμreal number-minusPprojectionλreal number) \( A\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\leq μ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{} \).

We now move towards the first version of the spectral theorem: Fix areal number<less thanbreal number \( a\lt b \) with σ(Aself-adjoint operator)subset(intervalareal number, breal number)interval \( σ\mathopen{}\left( A\right)\mathclose{}\subseteq \mathopen{}\left(a, b\right)\mathclose{} \). Consider a partition of [intervalareal number, breal number]interval \( \mathopen{}\left[a, b\right]\mathclose{} \): areal number=equalsλreal number0zero<less thanλreal number1one<less than<less thanλreal numberninteger-minus1one<less thanλreal numberninteger=equalsbreal number \( a= {λ}_{0}\lt {λ}_{1}\lt \dotsb\lt {λ}_{n-1}\lt {λ}_{n}= b \). Now pick treal numberjintegerelement of[intervalλreal numberjinteger-minus1one, λreal numberjinteger]interval \( {t}_{j}\in \mathopen{}\left[{λ}_{j-1}, {λ}_{j}\right]\mathclose{} \) for jintegerelement of{set1one2twoninteger}set \( j\in \mathopen{}\left\{\, 1, 2, \dotsc, n\,\right\}\mathclose{} \). Now consider Soperator=equalssummationjinteger=1oneninteger treal numberjintegertimes(Pprojectionλreal numberjinteger-minusPprojectionλreal numberjinteger-minus1one) \( S= \sum_{j=1}^{n}{} {t}_{j}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{} \).

We claim that Aself-adjoint operator-minusSoperatorless than or equal tomaxmaximumjinteger(λreal numberjinteger-minusλreal numberjinteger-minus1one) \( \mathopen{}\left\lVert{}A-S\right\rVert\mathclose{}\leq \max_{j}{}\mathopen{}\left({λ}_{j}-{λ}_{j-1}\right)\mathclose{} \). Let Qprojectionjinteger=equalsPprojectionλreal numberjinteger-minusPprojectionλreal numberjinteger-minus1one \( {Q}_{j}= {P}_{{λ}_{j}}-{P}_{{λ}_{j-1}} \). Note that summationjinteger=1onenintegerQprojectionjinteger=equals1one \( \sum_{j=1}^{n}{}{Q}_{j}= 1 \) and that the Qprojectionjinteger \( {Q}_{j} \)'s are mutually orthogonal: QprojectionjintegertimesQprojectionkinteger=equals0zero \( {Q}_{j}{Q}_{k}= 0 \) if jintegernot equal tokinteger \( j\neq k \). Then HHilbert space=equalsdirect product Qprojectionjinteger(HHilbert space) \( H= \bigoplus{} {Q}_{j}\mathopen{}\left( H\right)\mathclose{} \) so Aself-adjoint operator-minusSoperator=equalssummationjinteger=1onenintegerAself-adjoint operatortimesQprojectionjinteger-minussummationjinteger=1oneninteger treal numberjintegertimesQprojectionjinteger =equalssummationjinteger=1oneninteger (Aself-adjoint operator-minustreal numberjinteger)timesQprojectionjinteger \( A-S= \sum_{j=1}^{n}{}A{Q}_{j}-\sum_{j=1}^{n}{} {t}_{j}{Q}_{j} = \sum_{j=1}^{n}{} \mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j} \). Notice that for any operators Toperator1one \( {T}_{1} \), Toperator2two \( {T}_{2} \), …, Toperatorninteger \( {T}_{n} \) that commute with Qprojectionjinteger \( {Q}_{j} \), we have that summationjinteger=1oneninteger ToperatorjintegertimesQprojectionjinteger =equalsmaxmaximumjinteger ToperatorjintegertimesQprojectionjinteger \( \mathopen{}\left\lVert{}\sum_{j=1}^{n}{} {T}_{j}{Q}_{j} \right\rVert\mathclose{}= \max_{j}{} \mathopen{}\left\lVert{}{T}_{j}{Q}_{j}\right\rVert\mathclose{} \) because summationjinteger=1oneninteger ToperatorjintegertimesQprojectionjinteger greater than or equal tosummationjinteger=1oneninteger ToperatorjintegertimesQprojectionjinteger|restricted toQprojectionkinteger(HHilbert space) =equalsToperatorkintegertimesQprojectionkinteger|restricted toQprojectionkinteger(HHilbert space)=equalsToperatorkintegertimesQprojectionkinteger . \[ \mathopen{}\left\lVert{}\sum_{j=1}^{n}{} {T}_{j}{Q}_{j} \right\rVert\mathclose{}\geq \mathopen{}\left\lVert{}\sum_{j=1}^{n}{} {T}_{j}{Q}_{j}|{Q}_{k}\mathopen{}\left( H\right)\mathclose{} \right\rVert\mathclose{}= \mathopen{}\left\lVert{}{T}_{k}{Q}_{k}|{Q}_{k}\mathopen{}\left( H\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}{T}_{k}{Q}_{k}\right\rVert\mathclose{} \text{.} \]

For the proof of the reverse inequality, summationjinteger=1oneninteger Toperatorjinteger(Qprojectionjinteger(xvector)) 2two =equals summationjinteger=1oneninteger Qprojectionjinteger(Toperatorjinteger(Qprojectionjinteger(xvector))) 2two =equalssummationjinteger=1oneninteger Toperatorjinteger(Qprojectionjinteger(xvector)) 2two =equalssummationjinteger=1oneninteger Toperatorjinteger(Qprojectionjinteger(Qprojectionjinteger(xvector))) 2two less than or equal tosummationjinteger=1oneninteger Toperatorjinteger(Qprojectionjinteger) 2two times Qprojectionjinteger(xvector) 2two less than or equal tomaxmaximumkinteger Toperatorkinteger(Qprojectionkinteger) 2two timessummationjinteger=1oneninteger Qprojectionjinteger(xvector) 2two =equalsmaxmaximumkinteger Toperatorkinteger(Qprojectionkinteger) 2two timesxvector2two . \[ {\mathopen{}\left\lVert{}\sum_{j=1}^{n}{} {T}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{} \right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}\sum_{j=1}^{n}{} {Q}_{j}\mathopen{}\left( {T}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right)\mathclose{} \right\rVert\mathclose{}}^{2}= \sum_{j=1}^{n}{} {\mathopen{}\left\lVert{}{T}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right\rVert\mathclose{}}^{2} = \sum_{j=1}^{n}{} {\mathopen{}\left\lVert{}{T}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}\right\rVert\mathclose{}}^{2} \leq \sum_{j=1}^{n}{} {\mathopen{}\left\lVert{}{T}_{j}\mathopen{}\left( {Q}_{j}\right)\mathclose{}\right\rVert\mathclose{}}^{2}{\mathopen{}\left\lVert{}{Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2} \leq \max_{k}{} {\mathopen{}\left\lVert{}{T}_{k}\mathopen{}\left( {Q}_{k}\right)\mathclose{}\right\rVert\mathclose{}}^{2}\sum_{j=1}^{n}{} {\mathopen{}\left\lVert{}{Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2} = \max_{k}{} {\mathopen{}\left\lVert{}{T}_{k}\mathopen{}\left( {Q}_{k}\right)\mathclose{}\right\rVert\mathclose{}}^{2}{\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2} \text{.} \] So Aself-adjoint operator-minusSoperator=equalsmaxmaximumjinteger (Aself-adjoint operator-minustreal numberjinteger)timesQprojectionjinteger \( \mathopen{}\left\lVert{}A-S\right\rVert\mathclose{}= \max_{j}{} \mathopen{}\left\lVert{}\mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}\right\rVert\mathclose{} \). By Lemma IV.4, λreal numberjinteger-minus1onetimesQprojectionjintegerless than or equal toAself-adjoint operatortimesQprojectionjintegerless than or equal toλreal numberjintegertimesQprojectionjinteger \( {λ}_{j-1}{Q}_{j}\leq A{Q}_{j}\leq {λ}_{j}{Q}_{j} \). So (λreal numberjinteger-minus1one-minustreal numberjinteger)timesQprojectionjintegerless than or equal to(Aself-adjoint operator-minustreal numberjinteger)timesQprojectionjintegerless than or equal to(λreal numberjinteger-minustreal numberjinteger)timesQprojectionjinteger \( \mathopen{}\left({λ}_{j-1}-{t}_{j}\right)\mathclose{}{Q}_{j}\leq \mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}\leq \mathopen{}\left({λ}_{j}-{t}_{j}\right)\mathclose{}{Q}_{j} \) which makes (treal numberjinteger-minusλreal numberjinteger-minus1one) less than or equal to(Aself-adjoint operator-minustreal numberjinteger)timesQprojectionjintegerless than or equal to(λreal numberjinteger-minustreal numberjinteger) \( {-} \mathopen{}\left({t}_{j}-{λ}_{j-1}\right)\mathclose{} \leq \mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}\leq \mathopen{}\left({λ}_{j}-{t}_{j}\right)\mathclose{} \). We conclude that (Aself-adjoint operator-minustreal numberjinteger)timesQprojectionjintegerless than or equal tomaxmaximum (treal numberjinteger-minusλreal numberjinteger-minus1one, λreal numberjinteger-minustreal numberjinteger) less than or equal toλreal numberjinteger-minusλreal numberjinteger-minus1one \( \mathopen{}\left\lVert{}\mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}\right\rVert\mathclose{}\leq \max{} \mathopen{}\left({t}_{j}-{λ}_{j-1}, {λ}_{j}-{t}_{j}\right)\mathclose{} \leq {λ}_{j}-{λ}_{j-1} \). Thus Aself-adjoint operator-minusSoperatorless than or equal tomaxmaximum (λreal numberjinteger-minusλreal numberjinteger-minus1one) \( \mathopen{}\left\lVert{}A-S\right\rVert\mathclose{}\leq \max{} \mathopen{}\left({λ}_{j}-{λ}_{j-1}\right)\mathclose{} \).

We have now proved the spectral theorem for bounded self-adjoint operators. It remains to be seen what the spectral theorem has to do with the spectrum of such an operator.

Theorem IV.5

Given Aself-adjoint operator=equalsAself-adjoint operator*element ofbounded linear operators(HHilbert space) \( A= A^{*}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) and areal number<less thanbreal number \( a\lt b \) such that σ(areal number)subset(intervalareal number, breal number)interval \( σ\mathopen{}\left( a\right)\mathclose{}\subseteq \mathopen{}\left(a, b\right)\mathclose{} \) there is an increasing projection-valued function λreal numberis mapped toPprojectionλreal number \( λ\mapsto {P}_{λ} \) with Pprojectionareal number=equals0zero \( {P}_{a}= 0 \) and Pprojectionbreal number=equals1one \( {P}_{b}= 1 \) and each Pprojectionλreal number Aself-adjoint operator \( {P}_{λ} A \) such that Aself-adjoint operator=equalsintegralareal numberbreal numberλreal numberdPprojectionλreal number \( A= \int _{a}^{b}{}λ\,\mathrm{d}{P}_{λ} \) in the sense that Aself-adjoint operator\( A \) is the norm-limit of sums summationjinteger=1oneninteger treal numberjintegertimes(Pprojectionλreal numberjinteger-minusPprojectionλreal numberjinteger-minus1one) \( \sum_{j=1}^{n}{} {t}_{j}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{} \) over partitions areal number=equalsλreal number0zero<less thanλreal number1one<less than<less thanλreal numberninteger=equalsbreal number \( a= {λ}_{0}\lt {λ}_{1}\lt \dotsb\lt {λ}_{n}= b \) marked by treal numberjintegerelement of[intervalλreal numberjinteger-minus1one, λreal numberjinteger]interval \( {t}_{j}\in \mathopen{}\left[{λ}_{j-1}, {λ}_{j}\right]\mathclose{} \), as the mesh of the partition goes to 0zero\( 0 \).

Example IV.6

Mαcomplex numbermultiplication operatorαcomplex number\( \mathrm{M}_{α} \) is multiplication on L2Lebesgue space(Xnormed linear spaceμmeasure) \( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( X, μ\right)\mathclose{} \) by αcomplex numberelement ofLLebesgue space(Xnormed linear spaceμmeasure) \( α\in \mathrm{L}^{\mathrm{∞}}\mathopen{}\left( X, μ\right)\mathclose{} \), i.e. (Mαcomplex numbermultiplication operatorαcomplex number(ξfunction))(xelement)=equalsαcomplex number(xelement)timesξfunction(xelement) \( \mathopen{}\left(\mathrm{M}_{α}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= α\mathopen{}\left( x\right)\mathclose{}ξ\mathopen{}\left( x\right)\mathclose{} \). Mαcomplex numbermultiplication operatorαcomplex number-minusλreal number+plus|modulusMαcomplex numbermultiplication operatorαcomplex number-minusλreal number|modulus \( \mathrm{M}_{α}-λ+\mathopen{}\left\lvert{}\mathrm{M}_{α}-λ\right\rvert\mathclose{} \) is multiplication by αcomplex number-minusλreal number+plus|modulusαcomplex number-minusλreal number|modulus \( α-λ+\mathopen{}\left\lvert{}α-λ\right\rvert\mathclose{} \). Kerkernel(fcontinuous functionλreal number(Mαcomplex numbermultiplication operatorαcomplex number))=equalsL2Lebesgue space({setxelement|such that αcomplex number(xelement)less than or equal toλreal number }set) \( \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( \mathrm{M}_{α}\right)\mathclose{}\right)\mathclose{}= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left\{\, x\,\middle\vert\, , α\mathopen{}\left( x\right)\mathclose{}\leq λ, \,\right\}\mathclose{}\right)\mathclose{} \) (i.e. L2Lebesgue space(Xnormed linear space) \( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( X\right)\mathclose{} \) functions vanishing almost everywhere off {setxelement|such that αcomplex number(xelement)less than or equal toλreal number }set \( \mathopen{}\left\{\, x\,\middle\vert\, , α\mathopen{}\left( x\right)\mathclose{}\leq λ, \,\right\}\mathclose{} \)) and Pprojectionλreal number \( {P}_{λ} \) is projection on Kerkernel(fcontinuous functionλreal number(Mαcomplex numbermultiplication operatorαcomplex number)) \( \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( \mathrm{M}_{α}\right)\mathclose{}\right)\mathclose{} \).

In the setup of Theorem IV.5, consider gcontinuous functionelement ofC [intervalareal number, breal number]interval \( g\in \mathbb{C} \mathopen{}\left[a, b\right]\mathclose{} \). What is integralareal numberbreal numbergcontinuous function(λreal number)dPprojectionλreal number \( \int _{a}^{b}{}g\mathopen{}\left( λ\right)\mathclose{}\,\mathrm{d}{P}_{λ} \)? Given εpositive real number>greater than0zero \( ε\gt 0 \), get δpositive real number>greater than0zero \( δ\gt 0 \) such that |modulusgcontinuous function(sreal number)-minusgcontinuous function(treal number)|modulus<less thanεpositive real number \( \mathopen{}\left\lvert{}g\mathopen{}\left( s\right)\mathclose{}-g\mathopen{}\left( t\right)\mathclose{}\right\rvert\mathclose{}\lt ε \) whenever |modulussreal number-minustreal number|modulus<less thanδpositive real number \( \mathopen{}\left\lvert{}s-t\right\rvert\mathclose{}\lt δ \). Given a partition areal number=equalsλreal number0zero<less thanλreal number1one<less than<less thanλreal numberninteger=equalsbreal number \( a= {λ}_{0}\lt {λ}_{1}\lt \dotsb\lt {λ}_{n}= b \) with λreal numberjinteger-minusλreal numberjinteger-minus1one<less thanδpositive real number \( {λ}_{j}-{λ}_{j-1}\lt δ \) for all jinteger\( j \) and any refinement areal number=equalsμreal number0zero<less thanμreal number1one<less than<less thanμreal numberrreal number=equalsbreal number \( a= {μ}_{0}\lt {μ}_{1}\lt \dotsb\lt {μ}_{r}= b \). If we mark the λreal number\( λ \)-partition with treal numberjintegerelement of[intervalλreal numberjinteger-minus1one, λreal numberjinteger]interval \( {t}_{j}\in \mathopen{}\left[{λ}_{j-1}, {λ}_{j}\right]\mathclose{} \) and the μreal number\( μ \)-partition with sreal numberiintegerelement of[intervalμreal numberiinteger-minus1one, μreal numberiinteger]interval \( {s}_{i}\in \mathopen{}\left[{μ}_{i-1}, {μ}_{i}\right]\mathclose{} \) and form Toperator=equalssummationjinteger=1oneninteger gcontinuous function(treal numberjinteger)times(Pprojectionλreal numberjinteger-minusPprojectionλreal numberjinteger-minus1one) \( T= \sum_{j=1}^{n}{} g\mathopen{}\left( {t}_{j}\right)\mathclose{}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{} \) and Soperator=equalssummationiinteger=1onerreal number gcontinuous function(sreal numberiinteger)times(Pprojectionμreal numberiinteger-minusPprojectionμreal numberiinteger-minus1one) \( S= \sum_{i=1}^{r}{} g\mathopen{}\left( {s}_{i}\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{} \), then Soperator-minusToperator<less thanεpositive real number \( \mathopen{}\left\lVert{}S-T\right\rVert\mathclose{}\lt ε \) because everything commutes. Indeed, Soperator-minusToperator=equalssummationjinteger=1oneninteger (Soperator-minusToperator)times(Pprojectionλreal numberjinteger-minusPprojectionλreal numberjinteger-minus1one) =equalsmaxmaximumjinteger (Soperator-minusToperator)times(Pprojectionλreal numberjinteger-minusPprojectionλreal numberjinteger-minus1one) \[ \mathopen{}\left\lVert{}S-T\right\rVert\mathclose{}= \mathopen{}\left\lVert{}\sum_{j=1}^{n}{} \mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{} \right\rVert\mathclose{}= \max_{j}{} \mathopen{}\left\lVert{}\mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{}\right\rVert\mathclose{} \] and each of these is less than εpositive real number\( ε \) by the following calculation: Purely for convenience, let's take jinteger=equals1one \( j= 1 \). Write the refined partition of [intervalλreal number0zero, λreal number1one]interval \( \mathopen{}\left[{λ}_{0}, {λ}_{1}\right]\mathclose{} \) as λreal number0zero=equalsμreal number0zero<less thanμreal number1one<less than<less thanμreal numberkinteger=equalsλreal number1one \( {λ}_{0}= {μ}_{0}\lt {μ}_{1}\lt \dotsb\lt {μ}_{k}= {λ}_{1} \). Then (Soperator-minusToperator)times(Pprojectionλreal number1one-minusPprojectionλreal number0zero)=equalssummationiinteger=1onekinteger (Soperator-minusToperator)times(Pprojectionμreal numberiinteger-minusPprojectionμreal numberiinteger-minus1one) =equalsmaxmaximum1oneless than or equal toiintegerless than or equal tokinteger (Soperator-minusToperator)times(Pprojectionμreal numberiinteger-minusPprojectionμreal numberiinteger-minus1one) =equalsmaxmaximumiinteger (gcontinuous function(sreal numberiinteger)-minusgcontinuous function(treal number1one))times(Pprojectionμreal numberiinteger-minusPprojectionμreal numberiinteger-minus1one) less than or equal tomaxmaximumiinteger |modulusgcontinuous function(sreal numberiinteger)-minusgcontinuous function(treal number1one)|modulus <less thanεpositive real number , \[ \mathopen{}\left\lVert{}\mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{λ}_{1}}-{P}_{{λ}_{0}}\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}\sum_{i=1}^{k}{} \mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{} \right\rVert\mathclose{}= \max_{1\leq i\leq k}{} \mathopen{}\left\lVert{}\mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{}\right\rVert\mathclose{} = \max_{i}{} \mathopen{}\left\lVert{}\mathopen{}\left(g\mathopen{}\left( {s}_{i}\right)\mathclose{}-g\mathopen{}\left( {t}_{1}\right)\mathclose{}\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{}\right\rVert\mathclose{} \leq \max_{i}{} \mathopen{}\left\lvert{}g\mathopen{}\left( {s}_{i}\right)\mathclose{}-g\mathopen{}\left( {t}_{1}\right)\mathclose{}\right\rvert\mathclose{} \lt ε \text{,} \] because |modulussreal numberiinteger-minustreal number1one|modulus<less thanδpositive real number \( \mathopen{}\left\lvert{}{s}_{i}-{t}_{1}\right\rvert\mathclose{}\lt δ \) for these iinteger\( i \).

Remark IV.7

Because bounded linear operators(HHilbert space) \( \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) is complete it follows that there exists Toperatorgcontinuous functionelement ofbounded linear operators(HHilbert space) \( {T}_{g}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) —written Toperatorgcontinuous function=equalsintegralareal numberbreal numbergcontinuous function(λreal number)dPprojectionλreal number \( {T}_{g}= \int _{a}^{b}{}g\mathopen{}\left( λ\right)\mathclose{}\,\mathrm{d}{P}_{λ} \) —such that for all εpositive real number>greater than0zero \( ε\gt 0 \) there exists a partition Fpartition\( F \) such that Toperatorgcontinuous function-minusSoperator<less thanεpositive real number \( \mathopen{}\left\lVert{}{T}_{g}-S\right\rVert\mathclose{}\lt ε \) for any sum Soperator=equalssummation gcontinuous function(sreal numberiinteger)times(Pprojectionμreal numberiinteger-minusPprojectionμreal numberiinteger-minus1one) \( S= \sum{} g\mathopen{}\left( {s}_{i}\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{} \) in which {setμreal number0zeroμreal number1oneμreal numberninteger}set \( \mathopen{}\left\{\, {μ}_{0}, {μ}_{1}, \dotsc, {μ}_{n}\,\right\}\mathclose{} \) refines Fpartition\( F \).

Some easy properties:

  1. Toperatorgcontinuous function \( {T}_{g} \) commutes with Pprojectionλreal number \( {P}_{λ} \).
  2. Toperatorcreal numbertimesgcontinuous function1one+plusgcontinuous function2two=equalscreal numbertimesToperatorgcontinuous function1one+plusToperatorgcontinuous function2two \( {T}_{c{g}_{1}+{g}_{2}}= c{T}_{{g}_{1}}+{T}_{{g}_{2}} \).
  3. Toperatorgcontinuous function*=equalsToperatorgcontinuous function¯complex conjugate \( {T}_{g}^{*}= {T}_{\overline{g}} \).
  4. Toperatorgcontinuous function1onetimesgcontinuous function2two=equalsToperatorgcontinuous function1onetimesToperatorgcontinuous function2two \( {T}_{{g}_{1}{g}_{2}}= {T}_{{g}_{1}}{T}_{{g}_{2}} \).
  5. Toperator(gcontinuous function)less than or equal tomaxmaximum[intervalareal number, breal number]interval|modulusgcontinuous function|modulus \( \mathopen{}\left\lVert{}T\mathopen{}\left( g\right)\mathclose{}\right\rVert\mathclose{}\leq \max_{\mathopen{}\left[a, b\right]\mathclose{}}{}\mathopen{}\left\lvert{}g\right\rvert\mathclose{} \).

The only one of these that offers any resistance is item 4. For this, get a partition λreal number0zero<less thanλreal number1one<less than<less thanλreal numberninteger \( {λ}_{0}\lt {λ}_{1}\lt \dotsb\lt {λ}_{n} \) (by common refinement of a partition that works for gcontinuous function1one \( {g}_{1} \) and one that works for gcontinuous function2two \( {g}_{2} \)) such that summation gcontinuous function1(treal numberjinteger) Δdifference Pprojectionλreal numberjinteger \( \sum{} {g}_{1}\mathopen{}\left( {t}_{j}\right)\mathclose{} \, \Delta {P}_{{λ}_{j}} \) and summation gcontinuous function2(treal numberjinteger) Δdifference Pprojectionλreal numberjinteger \( \sum{} {g}_{2}\mathopen{}\left( {t}_{j}\right)\mathclose{} \, \Delta {P}_{{λ}_{j}} \) are close to Toperatorgcontinuous function1one \( {T}_{{g}_{1}} \) and Toperatorgcontinuous function2two \( {T}_{{g}_{2}} \) respectively (where ΔdifferencePprojectionλreal numberiinteger=equalsPprojectionλreal numberiinteger-minusPprojectionλreal numberiinteger-minus1one \( \Delta {P}_{{λ}_{i}}= {P}_{{λ}_{i}}-{P}_{{λ}_{i-1}} \)), and then (summation gcontinuous function1(treal numberjinteger) Δdifference Pprojectionλreal numberjinteger )times(summation gcontinuous function2(treal numberjinteger) Δdifference Pprojectionλreal numberjinteger )=equalssummation gcontinuous function1(treal numberjinteger)gcontinuous function2(treal numberjinteger) Δdifference Pprojectionλreal numberjinteger . \[ \mathopen{}\left(\sum{} {g}_{1}\mathopen{}\left( {t}_{j}\right)\mathclose{} \, \Delta {P}_{{λ}_{j}} \right)\mathclose{}\mathopen{}\left(\sum{} {g}_{2}\mathopen{}\left( {t}_{j}\right)\mathclose{} \, \Delta {P}_{{λ}_{j}} \right)\mathclose{}= \sum{} {g}_{1}\mathopen{}\left( {t}_{j}\right)\mathclose{}{g}_{2}\mathopen{}\left( {t}_{j}\right)\mathclose{} \, \Delta {P}_{{λ}_{j}} \text{.} \] since Pprojectionλreal numberjinteger times Pprojectionλreal numberkinteger =equals0zero \( {P}_{{λ}_{j}} {P}_{{λ}_{k}} = 0 \) for jintegernot equal tokinteger \( j\neq k \).

Theorem IV.8

Let Aself-adjoint operator=equalsAself-adjoint operator*element ofbounded linear operators(HHilbert space) \( A= A^{*}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) with σ(Aself-adjoint operator)subset(intervalareal number, breal number)interval \( σ\mathopen{}\left( A\right)\mathclose{}\subseteq \mathopen{}\left(a, b\right)\mathclose{} \) and spectral resolution λreal numberis mapped toPprojectionλreal number \( λ\mapsto {P}_{λ} \) as in Theorem IV.5. Then for gcontinuous functionelement ofCspace of continuous functions([intervalareal number, breal number]interval) \( g\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{} \) we have integralareal numberbreal number gcontinuous function(λreal number) dPprojectionλreal number=equals(gcontinuous function|restricted toσ(Aself-adjoint operator))(Aself-adjoint operator) \( \int _{a}^{b}{} g\mathopen{}\left( λ\right)\mathclose{} \,\mathrm{d}{P}_{λ}= \mathopen{}\left(g|σ\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( A\right)\mathclose{} \) where the left-hand side is the norm limit of Riemann-Stieltjes sums and the right-hand side comes from functional calculus on Cspace of continuous functions(σ(Aself-adjoint operator)) \( \mathrm{C}\mathopen{}\left( σ\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \).

Proof. We have shown that gcontinuous functionis mapped tointegralareal numberbreal number gcontinuous function(λreal number) dPprojectionλreal numberequivalentToperatorgcontinuous function \( g\mapsto \int _{a}^{b}{} g\mathopen{}\left( λ\right)\mathclose{} \,\mathrm{d}{P}_{λ}\equiv {T}_{g} \) is a *-homomorphism from Cspace of continuous functions([intervalareal number, breal number]interval) \( \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{} \) into bounded linear operators(HHilbert space) \( \mathcal{L}\mathopen{}\left( H\right)\mathclose{} \) with Toperatorgcontinuous functionless than or equal tomaxmaximum[intervalareal number, breal number]interval|modulusgcontinuous function|modulus \( \mathopen{}\left\lVert{}{T}_{g}\right\rVert\mathclose{}\leq \max_{\mathopen{}\left[a, b\right]\mathclose{}}{}\mathopen{}\left\lvert{}g\right\rvert\mathclose{} \). We have that integralareal numberbreal number 1one dPprojectionλreal number=equalsI \( \int _{a}^{b}{} 1 \,\mathrm{d}{P}_{λ}= I \) and integralareal numberbreal number λreal number dPprojectionλreal number=equalsAself-adjoint operator \( \int _{a}^{b}{} λ \,\mathrm{d}{P}_{λ}= A \), so for any polynomial ppolynomial(xvector) \( p\mathopen{}\left( x\right)\mathclose{} \) we have that integralareal numberbreal number ppolynomial(Aself-adjoint operator) dPprojectionλreal number=equalsppolynomial(Aself-adjoint operator) \( \int _{a}^{b}{} p\mathopen{}\left( A\right)\mathclose{} \,\mathrm{d}{P}_{λ}= p\mathopen{}\left( A\right)\mathclose{} \). Take gcontinuous functionelement ofCspace of continuous functions([intervalareal number, breal number]interval) \( g\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{} \), and write ĝ=equalsgcontinuous function|restricted toσ(Aself-adjoint operator) \( ĝ= g|σ\mathopen{}\left( A\right)\mathclose{} \). Get a sequence (sequence ppolynomialninteger )sequence \( \mathopen{}\left( {p}_{n} \right)\mathclose{} \) of polynomials with ppolynomialnintegerconverges togcontinuous function \( {p}_{n} \to g \) uniformly on [intervalareal number, breal number]interval \( \mathopen{}\left[a, b\right]\mathclose{} \). So ppolynomialninteger(Aself-adjoint operator)=equalsintegralareal numberbreal number ppolynomialninteger(λreal number) dPprojectionλreal numberconverges tointegralareal numberbreal number gcontinuous function(λreal number) dPprojectionλreal number \( {p}_{n}\mathopen{}\left( A\right)\mathclose{}= \int _{a}^{b}{} {p}_{n}\mathopen{}\left( λ\right)\mathclose{} \,\mathrm{d}{P}_{λ} \to \int _{a}^{b}{} g\mathopen{}\left( λ\right)\mathclose{} \,\mathrm{d}{P}_{λ} \) but also ppolynomialnintegerconverges toĝ \( {p}_{n} \to ĝ \) uniformly on σ(Aself-adjoint operator) \( σ\mathopen{}\left( A\right)\mathclose{} \), so ppolynomialninteger(Aself-adjoint operator)converges toĝ(Aself-adjoint operator) \( {p}_{n}\mathopen{}\left( A\right)\mathclose{} \to ĝ\mathopen{}\left( A\right)\mathclose{} \).

Remark IV.9

Theorem IV.8 says that for gcontinuous functionelement ofCspace of continuous functions([intervalareal number, breal number]interval) \( g\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{} \), the integral integralareal numberbreal number gcontinuous function(λreal number) dPprojectionλreal number \( \int _{a}^{b}{} g\mathopen{}\left( λ\right)\mathclose{} \,\mathrm{d}{P}_{λ} \) depends only on the restriction of gcontinuous function\( g \) to σ(Aself-adjoint operator) \( σ\mathopen{}\left( A\right)\mathclose{} \). It follows that an open real interval misses σ(Aself-adjoint operator) \( σ\mathopen{}\left( A\right)\mathclose{} \) if and only if λreal numberis mapped toPprojectionλreal number \( λ\mapsto {P}_{λ} \) is constant on that interval.

Eigenvalues of Aself-adjoint operator\( A \) show up in the spectral resolution as jump discontinuities, as we shall now explain.

Lemma IV.10

Let (sequenceEprojectionninteger)sequence \( \mathopen{}\left({E}_{n}\right)\mathclose{} \) be a decreasing sequence of projections, i.e. Eprojectionnintegergreater than or equal toEprojectionninteger+plus1one \( {E}_{n}\geq {E}_{n+1} \) for all ninteger\( n \), and let Eprojection\( E \) be the orthogonal projection on intersectionnintegerEprojectionninteger(HHilbert space) \( \bigcap_{n}{}{E}_{n}\mathopen{}\left( H\right)\mathclose{} \). Then Eprojectionninteger(ξvector)converges toEprojection(ξvector) \( {E}_{n}\mathopen{}\left( ξ\right)\mathclose{} \to E\mathopen{}\left( ξ\right)\mathclose{} \) for all ξvectorelement ofHHilbert space \( ξ\in H \).

Proof. The nonnegative sequence (Eprojectionninteger-minusEprojection)(ξvector) \( \mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{} \) is decreasing. Call the limit rreal number\( r \). Given εpositive real number>greater than0zero \( ε\gt 0 \), get Ninteger\( N \) such that (Eprojectionninteger-minusEprojection)(ξvector) 2two <less thanrreal number2two+plusεpositive real number \( {\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}\lt {r}^{2}+ε \) for nintegergreater than or equal toNinteger \( n\geq N \). If minteger>greater thannintegergreater than or equal toNinteger \( m\gt n\geq N \), then vectors (Eprojectionninteger-minusEprojectionminteger)(ξvector) \( \mathopen{}\left({E}_{n}-{E}_{m}\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{} \) and (Eprojectionminteger-minusEprojection)(ξvector) \( \mathopen{}\left({E}_{m}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{} \) are orthogonal, so (Eprojectionninteger-minusEprojection)(ξvector) 2two =equals (Eprojectionninteger-minusEprojectionminteger)(ξvector) 2two +plus (Eprojectionminteger-minusEprojection)(ξvector) 2two \[ {\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-{E}_{m}\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}+{\mathopen{}\left\lVert{}\mathopen{}\left({E}_{m}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2} \] and thus (Eprojectionninteger-minusEprojectionminteger)(ξvector) 2two \( {\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-{E}_{m}\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2} \) is the difference of two numbers in [intervalrreal number2two, rreal number2two+plusεpositive real number)interval \( \mathopen{}\left[{r}^{2}, {r}^{2}+ε\right)\mathclose{} \), which makes (Eprojectionninteger-minusEprojectionminteger)(ξvector) 2two <less thanεpositive real number \( {\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-{E}_{m}\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}\lt ε \). The sequence (sequence Eprojectionninteger(ξvector) )sequence \( \mathopen{}\left( {E}_{n}\mathopen{}\left( ξ\right)\mathclose{} \right)\mathclose{} \) therefore converges, say to ηvector\( η \). For any kinteger\( k \), we have Eprojectionkinteger(ηvector)=equalslimlimitninteger Eprojectionkinteger(Eprojectionninteger(ξvector)) =equalslimlimitninteger Eprojectionninteger(ξvector) =equalsηvector \( {E}_{k}\mathopen{}\left( η\right)\mathclose{}= \lim_{n}{} {E}_{k}\mathopen{}\left( {E}_{n}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{} = \lim_{n}{} {E}_{n}\mathopen{}\left( ξ\right)\mathclose{} = η \). Thus ηvector=equalsEprojection(ηvector)=equalslimlimitninteger Eprojection(Eprojectionninteger(ξvector)) =equalsEprojection(ξvector) \( η= E\mathopen{}\left( η\right)\mathclose{}= \lim_{n}{} E\mathopen{}\left( {E}_{n}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{} = E\mathopen{}\left( ξ\right)\mathclose{} \).

Remark IV.11

For λreal numberelement ofRreal numbers \( λ\in \mathbb{R} \), let Qprojectionλreal number \( {Q}_{λ} \) be the orthogonal projection on unionθreal number<less thanλreal number Pprojectionθreal number(HHilbert space) ¯complex conjugate \( \overline{ \bigcup_{θ\lt λ}{} {P}_{θ}\mathopen{}\left( H\right)\mathclose{} } \) and Rprojectionλreal number \( {R}_{λ} \) be the orthogonal projection on intersectionθreal number>greater thanλreal number Pprojectionθreal number(HHilbert space) \( \bigcap_{θ\gt λ}{} {P}_{θ}\mathopen{}\left( H\right)\mathclose{} \). We have Qprojectionλreal numberless than or equal toPprojectionλreal numberless than or equal toRprojectionλreal number \( {Q}_{λ}\leq {P}_{λ}\leq {R}_{λ} \). The lemma above yields limlimitθreal numberλreal number+ Pprojectionθreal number(ξvector) =equalsRprojectionλreal number(ξvector) \( \lim_{θ\to{λ}^{\mathrm{+}}}{} {P}_{θ}\mathopen{}\left( ξ\right)\mathclose{} = {R}_{λ}\mathopen{}\left( ξ\right)\mathclose{} \) for all ξvector\( ξ \), and likewise for the limit from below by taking orthogonal complements.

Proposition IV.12

For all λreal number\( λ \), Rprojectionλreal number-minusQprojectionλreal number \( {R}_{λ}-{Q}_{λ} \) is the orthogonal projection on Kerkernel(Aself-adjoint operator-minusλreal number) \( \operatorname{Ker}\mathopen{}\left( A-λ\right)\mathclose{} \). That is, λreal number\( λ \) is an eigenvalue of Aself-adjoint operator\( A \) if and only if Qprojectionλreal numbernot equal toRprojectionλreal number \( {Q}_{λ}\neq {R}_{λ} \).

Proof. For any ξvector\( ξ \), we have (Aself-adjoint operator-minusλreal number)2two(ξvector), ξvector=equalsintegralareal numberbreal number (θreal number-minusλreal number)2two d Pprojectionθreal number(ξvector), ξvector \( \mathopen{}\left\langle{}\mathopen{}\left\langle{}{\mathopen{}\left(A-λ\right)\mathclose{}}^{2}\mathopen{}\left( ξ\right)\mathclose{}\right\rangle\mathclose{}, ξ\right\rangle\mathclose{}= \int _{a}^{b}{} {\mathopen{}\left(θ-λ\right)\mathclose{}}^{2} \,\mathrm{d} \mathopen{}\left\langle{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}, ξ\right\rangle\mathclose{} \). If this scalar Riemann-Stieltjes integral vanishes, then Pprojectionθreal number(ξvector), ξvector \( \mathopen{}\left\langle{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}, ξ\right\rangle\mathclose{} \) (i.e., Pprojectionθreal number(ξvector) 2two \( {\mathopen{}\left\lVert{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2} \)) must be constant for θreal number>greater thanλreal number \( θ\gt λ \) and for θreal number<less thanλreal number \( θ\lt λ \), i.e. Pprojectionθreal number(ξvector) 2two =equalsξvector2two \( {\mathopen{}\left\lVert{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}ξ\right\rVert\mathclose{}}^{2} \) (and hence Pprojectionθreal number(ξvector)=equalsξvector \( {P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}= ξ \)) for θreal number>greater thanλreal number \( θ\gt λ \), and Pprojectionθreal number(ξvector)=equals0zero \( {P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}= 0 \) for θreal number<less thanλreal number \( θ\lt λ \). Thus, Rprojectionλreal number(ξvector)=equalsξvector \( {R}_{λ}\mathopen{}\left( ξ\right)\mathclose{}= ξ \) and Qprojectionλreal number(ξvector)=equals0zero \( {Q}_{λ}\mathopen{}\left( ξ\right)\mathclose{}= 0 \). We have shown that Kerkernel(Aself-adjoint operator-minusλreal number)subset(Rprojectionλreal number-minusQprojectionλreal number)(HHilbert space) \( \operatorname{Ker}\mathopen{}\left( A-λ\right)\mathclose{}\subseteq \mathopen{}\left({R}_{λ}-{Q}_{λ}\right)\mathclose{}\mathopen{}\left( H\right)\mathclose{} \). Conversely, if Rprojectionλreal number(ξvector)=equalsξvector \( {R}_{λ}\mathopen{}\left( ξ\right)\mathclose{}= ξ \) and Qprojectionλreal number(ξvector)=equals0zero \( {Q}_{λ}\mathopen{}\left( ξ\right)\mathclose{}= 0 \), then the function θreal numberis mapped toPprojectionθreal number(ξvector), ξvector \( θ\mapsto \mathopen{}\left\langle{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}, ξ\right\rangle\mathclose{} \) jumps from 0zero\( 0 \) to ξvector2two \( {\mathopen{}\left\lVert{}ξ\right\rVert\mathclose{}}^{2} \) across λreal number\( λ \) and is otherwise flat, so the value of the integral above is ξvector2two \( {\mathopen{}\left\lVert{}ξ\right\rVert\mathclose{}}^{2} \) times the value of the integrand at λreal number\( λ \). This is of course 0zero\( 0 \), making (Aself-adjoint operator-minusλreal number)(ξvector)=equals0zero \( \mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}= 0 \).

We turn now to unbounded (i.e. not necessarily bounded) operators, aiming ultimately at the spectral theorem for the densely defined self-adjoint ones. A natural example is the operator Toperator\( T \) with domain {setξfunctionelement ofL2Lebesgue space(Rreal numbers)|such that integralRreal numbers xvector2twotimes |modulusξfunction(xvector)|modulus 2two dxvector<less thaninfinity }set \( \mathopen{}\left\{\, ξ\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}\right)\mathclose{}\,\middle\vert\, , \int _{\mathbb{R}}{} {x}^{2}{\mathopen{}\left\lvert{}ξ\mathopen{}\left( x\right)\mathclose{}\right\rvert\mathclose{}}^{2} \,\mathrm{d}x\lt \infty, \,\right\}\mathclose{} \) defined by (Toperator(ξfunction))(xvector)=equalsxvectortimesξfunction(xvector) \( \mathopen{}\left(T\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= xξ\mathopen{}\left( x\right)\mathclose{} \).


Previous: Operators and Differential Equations back to top Next: Unbounded Operators