Lecture Notes in Functional Analysis
by William L. Paschke
edition 0.9
image/svg+xml
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
ℒ bounded linear operators ( H Hilbert space ) \(
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
Write
ℒ bounded linear operators ( H Hilbert space ) SA selfadjoint elements = equals { set A self-adjoint operator ∈ element of ℒ bounded linear operators ( H Hilbert space ) | such that A self-adjoint operator = equals A self-adjoint operator * } set \(
{
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
}_{\rm SA}= \mathopen{}\left\{\, A\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}\,\middle\vert\, A= A^{*}\,\right\}\mathclose{}
\) A self-adjoint operator \( A \) B self-adjoint operator \( B \)
ℒ bounded linear operators ( H Hilbert space ) SA selfadjoint elements \(
{
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
}_{\rm SA}
\)
A self-adjoint operator ≤ less than or equal to B self-adjoint operator \(
A\leq B
\)
B self-adjoint operator - minus A self-adjoint operator ≥ greater than or equal to 0 zero \(
B-A\geq 0
\)
〈 A self-adjoint operator ( x vector ) , x vector 〉 ≤ less than or equal to 〈 B self-adjoint operator ( x vector ) , x vector 〉 \(
\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}\leq \mathopen{}\left\langle{}B\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}
\)
x vector ∈ element of H Hilbert space \(
x\in H
\)
T operator ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
T\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
A self-adjoint operator ≤ less than or equal to B self-adjoint operator \(
A\leq B
\)
T operator * times A self-adjoint operator times T operator ≤ less than or equal to T operator * times B self-adjoint operator times T operator \(
T^{*}AT\leq T^{*}BT
\)
〈 T operator * ( A self-adjoint operator ( T operator ( x vector ) ) ) , x vector 〉 = equals 〈 A self-adjoint operator ( T operator ( x vector ) ) , T operator ( x vector ) 〉 ≤ less than or equal to 〈 B self-adjoint operator ( T operator ( x vector ) ) , x vector 〉 \(
\mathopen{}\left\langle{} T^{*}\mathopen{}\left( A\mathopen{}\left( T\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}, x\right\rangle\mathclose{}= \mathopen{}\left\langle{}A\mathopen{}\left( T\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, T\mathopen{}\left( x\right)\mathclose{}\right\rangle\mathclose{}\leq \mathopen{}\left\langle{}B\mathopen{}\left( T\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, x\right\rangle\mathclose{}
\)
x vector ∈ element of H Hilbert space \(
x\in H
\)
A self-adjoint operator ∈ element of
ℒ bounded linear operators ( H Hilbert space ) SA selfadjoint elements \(
A\in {
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
}_{\rm SA}
\)
f continuous function → converges to f continuous function ( A self-adjoint operator ) \(
f \to f\mathopen{}\left( A\right)\mathclose{}
\)
C space of continuous functions ( σ ( A self-adjoint operator ) ) \(
\mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
ℒ bounded linear operators ( H Hilbert space ) \(
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
f continuous function ≥ greater than or equal to 0 zero \(
f\geq 0
\)
σ ( A self-adjoint operator ) \(
\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}
\)
f continuous function ( A self-adjoint operator ) ≥ greater than or equal to 0 zero \(
f\mathopen{}\left( A\right)\mathclose{}\geq 0
\)
f continuous function ( A self-adjoint operator ) = equals
( f continuous function 1 one 2 two ( A self-adjoint operator ) ) 2 two = equals
( f continuous function 1 one 2 two ( A self-adjoint operator ) ) * times f continuous function 1 one 2 two ( A self-adjoint operator ) \(
f\mathopen{}\left( A\right)\mathclose{}= {\mathopen{}\left({f}^{\frac{1}{2}}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}}^{2}=
\mathopen{}\left({f}^{\frac{1}{2}}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
^{*}{f}^{\frac{1}{2}}\mathopen{}\left( A\right)\mathclose{}
\) f continuous function \( f \) g continuous function \( g \)
C space of continuous functions ( σ ( A self-adjoint operator ) R real numbers ) \(
\mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}, \mathbb{R}\right)\mathclose{}
\)
f continuous function ≤ less than or equal to g continuous function \(
f\leq g
\)
f continuous function ( A self-adjoint operator ) ≤ less than or equal to g continuous function ( A self-adjoint operator ) \(
f\mathopen{}\left( A\right)\mathclose{}\leq g\mathopen{}\left( A\right)\mathclose{}
\)
Definition IV.1
P projection ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
P\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
P projection = equals P projection * = equals P projection 2 two \(
P= P^{*}= {P}^{2}
\)
For a closed subspace
E closed subspace ⊆ subset H Hilbert space \(
E\subseteq H
\) H Hilbert space \( H \) E closed subspace \( E \) P projection \( P \)
P projection ( H Hilbert space ) \(
P\mathopen{}\left( H\right)\mathclose{}
\)
P projection : maps H Hilbert space → to P projection ( H Hilbert space ) \(
P : H \to P\mathopen{}\left( H\right)\mathclose{}
\) H Hilbert space \( H \)
P projection ( H Hilbert space ) \(
P\mathopen{}\left( H\right)\mathclose{}
\) P projection \( P \) Q projection \( Q \)
P projection ≤ less than or equal to Q projection \(
P\leq Q
\)
P projection ( H Hilbert space ) ⊆ subset Q projection ( H Hilbert space ) \(
P\mathopen{}\left( H\right)\mathclose{}\subseteq Q\mathopen{}\left( H\right)\mathclose{}
\)
Q projection times P projection = equals P projection \(
QP= P
\)
Q projection - minus P projection \(
Q-P
\)
Q projection ( H Hilbert space ) ∩ intersection
( P projection ( H Hilbert space ) ) ⊥ \(
Q\mathopen{}\left( H\right)\mathclose{}\cap {
\mathopen{}\left(P\mathopen{}\left( H\right)\mathclose{}\right)\mathclose{}
}^{\perp}
\)
Q projection ( H Hilbert space ) ∩ intersection ( 1 one - minus P projection ) ( H Hilbert space ) \(
Q\mathopen{}\left( H\right)\mathclose{}\cap \mathopen{}\left(1-P\right)\mathclose{}\mathopen{}\left( H\right)\mathclose{}
\)
Q projection ( H Hilbert space ) ⊖ P projection ( H Hilbert space ) \(
Q\mathopen{}\left( H\right)\mathclose{}\ominus P\mathopen{}\left( H\right)\mathclose{}
\)
Fix
A self-adjoint operator ∈ element of
ℒ bounded linear operators ( H Hilbert space ) SA selfadjoint elements \(
A\in {
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
}_{\rm SA}
\)
λ real number ∈ element of R real numbers \(
λ\in \mathbb{R}
\)
f continuous function λ real number \(
{f}_{λ}
\) R real numbers \( \mathbb{R} \)
f continuous function λ real number ( t real number ) = equals t real number - minus λ real number + plus | modulus t real number - minus λ real number | modulus \(
{f}_{λ}\mathopen{}\left( t\right)\mathclose{}= t-λ+\mathopen{}\left\lvert{}t-λ\right\rvert\mathclose{}
\)
f continuous function λ real number ∈ element of C space of continuous functions ( σ ( A self-adjoint operator ) ) \(
{f}_{λ}\in \mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
P projection λ real number \(
{P}_{λ}
\)
Ker kernel ( f continuous function λ real number ( A self-adjoint operator ) ) \(
\operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
Lemma IV.2
λ real number ≤ less than or equal to μ real number \(
λ\leq μ
\)
P projection λ real number ≤ less than or equal to P projection μ real number \(
{P}_{λ}\leq {P}_{μ}
\)
λ real number < less than min minimum σ ( A self-adjoint operator ) \(
λ\lt \min{}\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}
\)
P projection λ real number = equals 0 zero \(
{P}_{λ}= 0
\)
λ real number > greater than max maximum σ ( A self-adjoint operator ) \(
λ\gt \max{}\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}
\)
P projection λ real number = equals 1 one \(
{P}_{λ}= 1
\)
Notice
λ real number ≤ less than or equal to μ real number \(
λ\leq μ
\)
f continuous function μ real number = equals g continuous function times f continuous function λ real number \(
{f}_{μ}= g{f}_{λ}
\) g continuous function \( g \)
f continuous function μ real number ( A self-adjoint operator ) = equals g continuous function ( A self-adjoint operator ) times f continuous function λ real number ( A self-adjoint operator ) \(
{f}_{μ}\mathopen{}\left( A\right)\mathclose{}= g\mathopen{}\left( A\right)\mathclose{}{f}_{λ}\mathopen{}\left( A\right)\mathclose{}
\)
f continuous function λ real number ( A self-adjoint operator ) ( x vector ) = equals 0 zero \(
{f}_{λ}\mathopen{}\left( A\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= 0
\)
f continuous function μ real number ( A self-adjoint operator ) ( x vector ) = equals 0 zero \(
{f}_{μ}\mathopen{}\left( A\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= 0
\)
Ker kernel ( f continuous function λ real number ( A self-adjoint operator ) ) ⊆ subset Ker kernel ( f continuous function μ real number ( A self-adjoint operator ) ) \(
\operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}\subseteq \operatorname{Ker}\mathopen{}\left( {f}_{μ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
λ real number < less than min minimum σ ( A self-adjoint operator ) \(
λ\lt \min{}\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}
\)
A self-adjoint operator - minus λ real number ≥ greater than or equal to 0 zero \(
A-λ\geq 0
\)
A self-adjoint operator - minus λ real number \(
A-λ
\)
f continuous function λ real number ( A self-adjoint operator ) = equals 2 two times ( A self-adjoint operator - minus λ real number ) \(
{f}_{λ}\mathopen{}\left( A\right)\mathclose{}= 2\mathopen{}\left(A-λ\right)\mathclose{}
\)
P projection λ real number = equals 0 zero \(
{P}_{λ}= 0
\)
λ real number > greater than max maximum σ ( A self-adjoint operator ) \(
λ\gt \max{}\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}
\)
A self-adjoint operator - minus λ real number ≤ less than or equal to 0 zero \(
A-λ\leq 0
\)
| modulus A self-adjoint operator - minus λ real number | modulus = equals λ real number - minus A self-adjoint operator \(
\mathopen{}\left\lvert{}A-λ\right\rvert\mathclose{}= λ-A
\)
f continuous function λ real number ( A self-adjoint operator ) = equals 0 zero \(
{f}_{λ}\mathopen{}\left( A\right)\mathclose{}= 0
\)
Ker kernel ( f continuous function λ real number ( A self-adjoint operator ) ) = equals H Hilbert space \(
\operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}= H
\)
Lemma IV.3
A self-adjoint operator times P projection λ real number = equals P projection λ real number times A self-adjoint operator \(
A{P}_{λ}= {P}_{λ}A
\)
A self-adjoint operator times P projection λ real number ≤ less than or equal to λ real number times P projection λ real number \(
A{P}_{λ}\leq λ{P}_{λ}
\)
A self-adjoint operator times ( 1 one - minus P projection λ real number ) ≥ greater than or equal to λ real number times ( 1 one - minus P projection λ real number ) \(
A\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\geq λ\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}
\)
Let
x vector ∈ element of Ker kernel ( f continuous function λ real number ( A self-adjoint operator ) ) \(
x\in \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
f continuous function λ real number ( A self-adjoint operator ) ( A self-adjoint operator ( x vector ) ) = equals A self-adjoint operator ( f continuous function λ real number ( A self-adjoint operator ( x vector ) ) ) = equals 0 zero \(
{f}_{λ}\mathopen{}\left( A\right)\mathclose{}\mathopen{}\left( A\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}= A\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}= 0
\)
A self-adjoint operator times Ker kernel ( f continuous function λ real number ( A self-adjoint operator ) ) ⊆ subset Ker kernel ( f continuous function λ real number ( A self-adjoint operator ) ) \(
A\operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}\subseteq \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
P projection λ real number times A self-adjoint operator times P projection λ real number = equals A self-adjoint operator times P projection λ real number \(
{P}_{λ}A{P}_{λ}= A{P}_{λ}
\)
P projection λ real number times A self-adjoint operator = equals P projection λ real number * times A self-adjoint operator * = equals ( P projection λ real number times A self-adjoint operator times P projection λ real number ) * = equals P projection λ real number * times A self-adjoint operator * times P projection λ real number * = equals P projection λ real number times A self-adjoint operator times P projection λ real number = equals A self-adjoint operator times P projection λ real number . \[
{P}_{λ}A= {P}_{λ}^{*} A^{*}= \mathopen{}\left({P}_{λ}A{P}_{λ}\right)\mathclose{}^{*}= {P}_{λ}^{*} A^{*} {P}_{λ}^{*}= {P}_{λ}A{P}_{λ}= A{P}_{λ}
\text{.}
\]
For
y vector ∈ element of Ker kernel ( f continuous function λ real number ( A self-adjoint operator ) ) \(
y\in \operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
( A self-adjoint operator - minus λ real number ) times y vector = equals − | modulus A self-adjoint operator - minus λ real number | modulus times y vector \(
\mathopen{}\left(A-λ\right)\mathclose{}y= {-}\mathopen{}\left\lvert{}A-λ\right\rvert\mathclose{}y
\)
x vector ∈ element of H Hilbert space \(
x\in H
\)
〈 ( A self-adjoint operator - minus λ real number ) ( P projection λ real number ( x vector ) ) , x vector 〉 = equals 〈 P projection λ real number ( ( A self-adjoint operator - minus λ real number ) ( P projection λ real number ( x vector ) ) x vector ) 〉 = equals 〈 ( A self-adjoint operator - minus λ real number ) ( P projection λ real number ( x vector ) ) , P projection λ real number ( x vector ) 〉 = equals −
〈 | modulus A self-adjoint operator - minus λ real number | modulus ( P projection λ real number ( x vector ) ) , P projection λ real number ( x vector ) 〉 ≤ less than or equal to 0 zero , \[
\mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, x\right\rangle\mathclose{}= \mathopen{}\left\langle{}{P}_{λ}\mathopen{}\left( \mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, x\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right\rangle\mathclose{}= {-}
\mathopen{}\left\langle{}\mathopen{}\left\lvert{}A-λ\right\rvert\mathclose{}\mathopen{}\left( {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}, {P}_{λ}\mathopen{}\left( x\right)\mathclose{}\right\rangle\mathclose{}
\leq 0
\text{,}
\]
( A self-adjoint operator - minus λ real number ) ( P projection λ real number ) ≤ less than or equal to 0 zero \(
\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {P}_{λ}\right)\mathclose{}\leq 0
\)
For any
y vector ∈ element of H Hilbert space \(
y\in H
\)
〈 ( A self-adjoint operator - minus λ real number ) ( f continuous function λ real number ( A self-adjoint operator 1 one 2 two ( y vector ) ) ) , f continuous function λ real number ( A self-adjoint operator 1 one 2 two ( y vector ) ) 〉 = equals 〈 ( A self-adjoint operator - minus λ real number ) ( f continuous function λ real number ( A self-adjoint operator ( y vector ) ) ) , y vector 〉 = equals 〈 (
( A self-adjoint operator - minus λ real number ) 2 two + plus ( A self-adjoint operator - minus λ real number ) times | modulus A self-adjoint operator - minus λ real number | modulus ) ( y vector ) , y vector 〉 ≥ greater than or equal to 0 zero . \[
\mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}, {f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( {f}_{λ}\mathopen{}\left( A\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}, y\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left({\mathopen{}\left(A-λ\right)\mathclose{}}^{2}+\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left\lvert{}A-λ\right\rvert\mathclose{}\right)\mathclose{}\mathopen{}\left( y\right)\mathclose{}, y\right\rangle\mathclose{}\geq 0
\text{.}
\]
〈 ( A self-adjoint operator - minus λ real number ) ( x vector ) , x vector 〉 ≥ greater than or equal to 0 zero \(
\mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}\geq 0
\)
x vector ∈ element of
f continuous function λ real number ( A self-adjoint operator 1 one 2 two ( H Hilbert space ) ) ¯ complex conjugate \(
x\in \overline{
{f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( H\right)\mathclose{}\right)\mathclose{}
}
\)
f continuous function λ real number ( A self-adjoint operator 1 one 2 two ( H Hilbert space ) ) ¯ complex conjugate = equals
( Ker kernel (
( f continuous function λ real number ( A self-adjoint operator 1 one 2 two ) ) * ) ) ⊥ = equals Ker kernel (
( f continuous function λ real number ( A self-adjoint operator 1 one 2 two ) ) ⊥ ) = equals ( 1 one - minus P projection λ real number ) ( H Hilbert space ) , \[
\overline{
{f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\mathopen{}\left( H\right)\mathclose{}\right)\mathclose{}
}= {
\mathopen{}\left(\operatorname{Ker}\mathopen{}\left(
\mathopen{}\left({f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\right)\mathclose{}\right)\mathclose{}
^{*}\right)\mathclose{}\right)\mathclose{}
}^{\perp}= \operatorname{Ker}\mathopen{}\left( {
\mathopen{}\left({f}_{λ}\mathopen{}\left( {A}^{\frac{1}{2}}\right)\mathclose{}\right)\mathclose{}
}^{\perp}\right)\mathclose{}= \mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left( H\right)\mathclose{}
\text{,}
\]
0 zero ≤ less than or equal to 〈 ( A self-adjoint operator - minus λ real number ) ( ( 1 one - minus P projection λ real number ) ( y vector ) ) , ( 1 one - minus P projection λ real number ) ( y vector ) 〉 = equals 〈 ( A self-adjoint operator - minus λ real number ) ( ( 1 one - minus P projection λ real number ) ( y vector ) ) , y vector 〉 . \[
0\leq \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( \mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}, \mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{}\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( \mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left( y\right)\mathclose{}\right)\mathclose{}, y\right\rangle\mathclose{}
\text{.}
\]
Lemma IV.4
For
λ real number ≤ less than or equal to μ real number \(
λ\leq μ
\)
λ real number times ( P projection μ real number - minus P projection λ real number ) ≤ less than or equal to A self-adjoint operator times ( P projection μ real number - minus P projection λ real number ) ≤ less than or equal to μ real number times ( P projection μ real number - minus P projection λ real number ) \(
λ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\leq A\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\leq μ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}
\)
Continuing with the proof of the spectral theorem (which we will state at the end), fix
a real number < less than b real number \(
a\lt b
\)
σ ( A self-adjoint operator ) ⊆ subset ( interval a real number , b real number ) interval \(
\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\subseteq \mathopen{}\left(a, b\right)\mathclose{}
\)
( P projection μ real number - minus P projection λ real number ) times A self-adjoint operator times ( 1 one - minus P projection λ real number ) times ( P projection μ real number - minus P projection λ real number ) ≥ greater than or equal to λ real number times ( P projection μ real number - minus P projection λ real number ) times ( 1 one - minus P projection λ real number ) times ( P projection μ real number - minus P projection λ real number ) \[
\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}A\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\geq λ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}
\]
A self-adjoint operator times ( 1 one - minus P projection λ real number ) times ( P projection μ real number - minus P projection λ real number ) ≥ greater than or equal to λ real number times ( 1 one - minus P projection λ real number ) times ( P projection μ real number - minus P projection λ real number ) = equals λ real number times ( P projection μ real number - minus P projection λ real number times P projection μ real number - minus P projection λ real number + plus P projection λ real number ) = equals λ real number times ( P projection μ real number - minus P projection λ real number ) \[
A\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\geq λ\mathopen{}\left(1-{P}_{λ}\right)\mathclose{}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}= λ\mathopen{}\left({P}_{μ}-{P}_{λ}{P}_{μ}-{P}_{λ}+{P}_{λ}\right)\mathclose{}= λ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}
\]
P projection λ real number ≤ less than or equal to P projection μ real number \(
{P}_{λ}\leq {P}_{μ}
\)
A self-adjoint operator times P projection μ real number - minus P projection λ real number ≥ greater than or equal to λ real number times ( P projection μ real number - minus P projection λ real number ) \(
A{P}_{μ}-{P}_{λ}\geq λ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}
\)
A self-adjoint operator times P projection μ real number ≤ less than or equal to μ real number times P projection μ real number \(
A{P}_{μ}\leq μ{P}_{μ}
\)
( P projection μ real number - minus P projection λ real number ) times A self-adjoint operator times P projection μ real number times ( P projection μ real number - minus P projection λ real number ) ≤ less than or equal to μ real number times ( P projection μ real number - minus P projection λ real number ) times P projection μ real number times ( P projection μ real number - minus P projection λ real number ) \[
\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}A{P}_{μ}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}\leq μ\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}{P}_{μ}\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}
\]
A self-adjoint operator times
( P projection μ real number - minus P projection λ real number ) 2 two ≤ less than or equal to μ real number times
( P projection μ real number - minus P projection λ real number ) 2 two \(
A{\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}}^{2}\leq μ{\mathopen{}\left({P}_{μ}-{P}_{λ}\right)\mathclose{}}^{2}
\)
A self-adjoint operator times ( P projection μ real number - minus P projection λ real number ) ≤ less than or equal to μ real number times ( P projection μ real number - minus P projection λ real 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
a real number < less than b real number \(
a\lt b
\)
σ ( A self-adjoint operator ) ⊆ subset ( interval a real number , b real number ) interval \(
σ\mathopen{}\left( A\right)\mathclose{}\subseteq \mathopen{}\left(a, b\right)\mathclose{}
\)
[ interval a real number , b real number ] interval \(
\mathopen{}\left[a, b\right]\mathclose{}
\)
a real number = equals λ real number 0 zero < less than λ real number 1 one < less than ⋯ < less than λ real number n integer - minus 1 one < less than λ real number n integer = equals b real number \(
a= {λ}_{0}\lt {λ}_{1}\lt \dotsb\lt {λ}_{n-1}\lt {λ}_{n}= b
\)
t real number j integer ∈ element of [ interval λ real number j integer - minus 1 one , λ real number j integer ] interval \(
{t}_{j}\in \mathopen{}\left[{λ}_{j-1}, {λ}_{j}\right]\mathclose{}
\)
j integer ∈ element of { set 1 one 2 two … n integer } set \(
j\in \mathopen{}\left\{\, 1, 2, \dotsc, n\,\right\}\mathclose{}
\)
S operator = equals ∑ summation j integer = 1 one n integer
t real number j integer times ( P projection λ real number j integer - minus P projection λ real number j integer - minus 1 one ) \(
S= \sum_{j=1}^{n}{}
{t}_{j}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{}
\)
We claim that
‖ A self-adjoint operator - minus S operator ‖ ≤ less than or equal to max maximum j integer ( λ real number j integer - minus λ real number j integer - minus 1 one ) \(
\mathopen{}\left\lVert{}A-S\right\rVert\mathclose{}\leq \max_{j}{}\mathopen{}\left({λ}_{j}-{λ}_{j-1}\right)\mathclose{}
\)
Q projection j integer = equals P projection λ real number j integer - minus P projection λ real number j integer - minus 1 one \(
{Q}_{j}= {P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}
\)
∑ summation j integer = 1 one n integer Q projection j integer = equals 1 one \(
\sum_{j=1}^{n}{}{Q}_{j}= 1
\)
Q projection j integer \(
{Q}_{j}
\)
Q projection j integer times Q projection k integer = equals 0 zero \(
{Q}_{j}{Q}_{k}= 0
\)
j integer ≠ not equal to k integer \(
j\neq k
\)
H Hilbert space = equals ⨁ direct product
Q projection j integer ( H Hilbert space ) \(
H= \bigoplus{}
{Q}_{j}\mathopen{}\left( H\right)\mathclose{}
\)
A self-adjoint operator - minus S operator = equals ∑ summation j integer = 1 one n integer A self-adjoint operator times Q projection j integer - minus ∑ summation j integer = 1 one n integer
t real number j integer times Q projection j integer = equals ∑ summation j integer = 1 one n integer
( A self-adjoint operator - minus t real number j integer ) times Q projection j integer \(
A-S= \sum_{j=1}^{n}{}A{Q}_{j}-\sum_{j=1}^{n}{}
{t}_{j}{Q}_{j}
= \sum_{j=1}^{n}{}
\mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}
\)
T operator 1 one \(
{T}_{1}
\)
T operator 2 two \(
{T}_{2}
\)
T operator n integer \(
{T}_{n}
\)
Q projection j integer \(
{Q}_{j}
\)
‖ ∑ summation j integer = 1 one n integer
T operator j integer times Q projection j integer ‖ = equals max maximum j integer
‖ T operator j integer times Q projection j integer ‖ \(
\mathopen{}\left\lVert{}\sum_{j=1}^{n}{}
{T}_{j}{Q}_{j}
\right\rVert\mathclose{}= \max_{j}{}
\mathopen{}\left\lVert{}{T}_{j}{Q}_{j}\right\rVert\mathclose{}
\)
‖ ∑ summation j integer = 1 one n integer
T operator j integer times Q projection j integer ‖ ≥ greater than or equal to ‖ ∑ summation j integer = 1 one n integer
T operator j integer times Q projection j integer | restricted to Q projection k integer ( H Hilbert space ) ‖ = equals ‖ T operator k integer times Q projection k integer | restricted to Q projection k integer ( H Hilbert space ) ‖ = equals ‖ T operator k integer times Q projection k integer ‖ . \[
\mathopen{}\left\lVert{}\sum_{j=1}^{n}{}
{T}_{j}{Q}_{j}
\right\rVert\mathclose{}\geq \mathopen{}\left\lVert{}\sum_{j=1}^{n}{}
{T}_{j}{Q}_{j}|{Q}_{k}\mathopen{}\left( H\right)\mathclose{}
\right\rVert\mathclose{}= \mathopen{}\left\lVert{}{T}_{k}{Q}_{k}|{Q}_{k}\mathopen{}\left( H\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}{T}_{k}{Q}_{k}\right\rVert\mathclose{}
\text{.}
\]
For the proof of the reverse inequality,
‖ ∑ summation j integer = 1 one n integer
T operator j integer ( Q projection j integer ( x vector ) ) ‖ 2 two = equals
‖ ∑ summation j integer = 1 one n integer
Q projection j integer ( T operator j integer ( Q projection j integer ( x vector ) ) ) ‖ 2 two = equals ∑ summation j integer = 1 one n integer
‖ T operator j integer ( Q projection j integer ( x vector ) ) ‖ 2 two = equals ∑ summation j integer = 1 one n integer
‖ T operator j integer ( Q projection j integer ( Q projection j integer ( x vector ) ) ) ‖ 2 two ≤ less than or equal to ∑ summation j integer = 1 one n integer
‖ T operator j integer ( Q projection j integer ) ‖ 2 two times
‖ Q projection j integer ( x vector ) ‖ 2 two ≤ less than or equal to max maximum k integer
‖ T operator k integer ( Q projection k integer ) ‖ 2 two times ∑ summation j integer = 1 one n integer
‖ Q projection j integer ( x vector ) ‖ 2 two = equals max maximum k integer
‖ T operator k integer ( Q projection k integer ) ‖ 2 two times ‖ x vector ‖ 2 two . \[
{\mathopen{}\left\lVert{}\sum_{j=1}^{n}{}
{T}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}
\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}\sum_{j=1}^{n}{}
{Q}_{j}\mathopen{}\left( {T}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}
\right\rVert\mathclose{}}^{2}= \sum_{j=1}^{n}{}
{\mathopen{}\left\lVert{}{T}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right\rVert\mathclose{}}^{2}
= \sum_{j=1}^{n}{}
{\mathopen{}\left\lVert{}{T}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( {Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}\right\rVert\mathclose{}}^{2}
\leq \sum_{j=1}^{n}{}
{\mathopen{}\left\lVert{}{T}_{j}\mathopen{}\left( {Q}_{j}\right)\mathclose{}\right\rVert\mathclose{}}^{2}{\mathopen{}\left\lVert{}{Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}
\leq \max_{k}{}
{\mathopen{}\left\lVert{}{T}_{k}\mathopen{}\left( {Q}_{k}\right)\mathclose{}\right\rVert\mathclose{}}^{2}\sum_{j=1}^{n}{}
{\mathopen{}\left\lVert{}{Q}_{j}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}}^{2}
= \max_{k}{}
{\mathopen{}\left\lVert{}{T}_{k}\mathopen{}\left( {Q}_{k}\right)\mathclose{}\right\rVert\mathclose{}}^{2}{\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}
\text{.}
\]
‖ A self-adjoint operator - minus S operator ‖ = equals max maximum j integer
‖ ( A self-adjoint operator - minus t real number j integer ) times Q projection j integer ‖ \(
\mathopen{}\left\lVert{}A-S\right\rVert\mathclose{}= \max_{j}{}
\mathopen{}\left\lVert{}\mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}\right\rVert\mathclose{}
\) Lemma IV.4 ,
λ real number j integer - minus 1 one times Q projection j integer ≤ less than or equal to A self-adjoint operator times Q projection j integer ≤ less than or equal to λ real number j integer times Q projection j integer \(
{λ}_{j-1}{Q}_{j}\leq A{Q}_{j}\leq {λ}_{j}{Q}_{j}
\)
( λ real number j integer - minus 1 one - minus t real number j integer ) times Q projection j integer ≤ less than or equal to ( A self-adjoint operator - minus t real number j integer ) times Q projection j integer ≤ less than or equal to ( λ real number j integer - minus t real number j integer ) times Q projection j integer \(
\mathopen{}\left({λ}_{j-1}-{t}_{j}\right)\mathclose{}{Q}_{j}\leq \mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}\leq \mathopen{}\left({λ}_{j}-{t}_{j}\right)\mathclose{}{Q}_{j}
\)
−
( t real number j integer - minus λ real number j integer - minus 1 one ) ≤ less than or equal to ( A self-adjoint operator - minus t real number j integer ) times Q projection j integer ≤ less than or equal to ( λ real number j integer - minus t real number j integer ) \(
{-}
\mathopen{}\left({t}_{j}-{λ}_{j-1}\right)\mathclose{}
\leq \mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}\leq \mathopen{}\left({λ}_{j}-{t}_{j}\right)\mathclose{}
\)
‖ ( A self-adjoint operator - minus t real number j integer ) times Q projection j integer ‖ ≤ less than or equal to max maximum
( t real number j integer - minus λ real number j integer - minus 1 one , λ real number j integer - minus t real number j integer ) ≤ less than or equal to λ real number j integer - minus λ real number j integer - minus 1 one \(
\mathopen{}\left\lVert{}\mathopen{}\left(A-{t}_{j}\right)\mathclose{}{Q}_{j}\right\rVert\mathclose{}\leq \max{}
\mathopen{}\left({t}_{j}-{λ}_{j-1}, {λ}_{j}-{t}_{j}\right)\mathclose{}
\leq {λ}_{j}-{λ}_{j-1}
\)
‖ A self-adjoint operator - minus S operator ‖ ≤ less than or equal to max maximum
( λ real number j integer - minus λ real number j integer - minus 1 one ) \(
\mathopen{}\left\lVert{}A-S\right\rVert\mathclose{}\leq \max{}
\mathopen{}\left({λ}_{j}-{λ}_{j-1}\right)\mathclose{}
\)
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
A self-adjoint operator = equals A self-adjoint operator * ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
A= A^{*}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
a real number < less than b real number \(
a\lt b
\)
σ ( a real number ) ⊆ subset ( interval a real number , b real number ) interval \(
σ\mathopen{}\left( a\right)\mathclose{}\subseteq \mathopen{}\left(a, b\right)\mathclose{}
\)
λ real number ↦ is mapped to P projection λ real number \(
λ\mapsto {P}_{λ}
\)
P projection a real number = equals 0 zero \(
{P}_{a}= 0
\)
P projection b real number = equals 1 one \(
{P}_{b}= 1
\)
P projection λ real number A self-adjoint operator \(
{P}_{λ}
A
\)
A self-adjoint operator = equals ∫ integral a real number b real number λ real number d P projection λ real number \(
A= \int _{a}^{b}{}λ\,\mathrm{d}{P}_{λ}
\) A self-adjoint operator \( A \)
∑ summation j integer = 1 one n integer
t real number j integer times ( P projection λ real number j integer - minus P projection λ real number j integer - minus 1 one ) \(
\sum_{j=1}^{n}{}
{t}_{j}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{}
\)
a real number = equals λ real number 0 zero < less than λ real number 1 one < less than ⋯ < less than λ real number n integer = equals b real number \(
a= {λ}_{0}\lt {λ}_{1}\lt \dotsb\lt {λ}_{n}= b
\)
t real number j integer ∈ element of [ interval λ real number j integer - minus 1 one , λ real number j integer ] interval \(
{t}_{j}\in \mathopen{}\left[{λ}_{j-1}, {λ}_{j}\right]\mathclose{}
\) 0 zero \( 0 \)
Example IV.6
M α complex number multiplication operator α complex number \( \mathrm{M}_{α} \)
L 2 Lebesgue space ( X normed linear space μ measure ) \(
\mathrm{L}^{\mathrm{2}}\mathopen{}\left( X, μ\right)\mathclose{}
\)
α complex number ∈ element of L ∞ Lebesgue space ( X normed linear space μ measure ) \(
α\in \mathrm{L}^{\mathrm{∞}}\mathopen{}\left( X, μ\right)\mathclose{}
\)
( M α complex number multiplication operator α complex number ( ξ function ) ) ( x element ) = equals α complex number ( x element ) times ξ function ( x element ) \(
\mathopen{}\left(\mathrm{M}_{α}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= α\mathopen{}\left( x\right)\mathclose{}ξ\mathopen{}\left( x\right)\mathclose{}
\)
M α complex number multiplication operator α complex number - minus λ real number + plus | modulus M α complex number multiplication operator α complex number - minus λ real number | modulus \(
\mathrm{M}_{α}-λ+\mathopen{}\left\lvert{}\mathrm{M}_{α}-λ\right\rvert\mathclose{}
\)
α complex number - minus λ real number + plus | modulus α complex number - minus λ real number | modulus \(
α-λ+\mathopen{}\left\lvert{}α-λ\right\rvert\mathclose{}
\)
Ker kernel ( f continuous function λ real number ( M α complex number multiplication operator α complex number ) ) = equals L 2 Lebesgue space ( { set x element | such that
α complex number ( x element ) ≤ less than or equal to λ real number } set ) \(
\operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( \mathrm{M}_{α}\right)\mathclose{}\right)\mathclose{}= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left\{\, x\,\middle\vert\,
, α\mathopen{}\left( x\right)\mathclose{}\leq λ,
\,\right\}\mathclose{}\right)\mathclose{}
\)
L 2 Lebesgue space ( X normed linear space ) \(
\mathrm{L}^{\mathrm{2}}\mathopen{}\left( X\right)\mathclose{}
\)
{ set x element | such that
α complex number ( x element ) ≤ less than or equal to λ real number } set \(
\mathopen{}\left\{\, x\,\middle\vert\,
, α\mathopen{}\left( x\right)\mathclose{}\leq λ,
\,\right\}\mathclose{}
\)
P projection λ real number \(
{P}_{λ}
\)
Ker kernel ( f continuous function λ real number ( M α complex number multiplication operator α complex number ) ) \(
\operatorname{Ker}\mathopen{}\left( {f}_{λ}\mathopen{}\left( \mathrm{M}_{α}\right)\mathclose{}\right)\mathclose{}
\)
In the setup of Theorem IV.5 , consider
g continuous function ∈ element of C [ interval a real number , b real number ] interval \(
g\in \mathbb{C}
\mathopen{}\left[a, b\right]\mathclose{}
\)
∫ integral a real number b real number g continuous function ( λ real number ) d P projection λ real number \(
\int _{a}^{b}{}g\mathopen{}\left( λ\right)\mathclose{}\,\mathrm{d}{P}_{λ}
\)
ε positive real number > greater than 0 zero \(
ε\gt 0
\)
δ positive real number > greater than 0 zero \(
δ\gt 0
\)
| modulus g continuous function ( s real number ) - minus g continuous function ( t real number ) | modulus < less than ε positive real number \(
\mathopen{}\left\lvert{}g\mathopen{}\left( s\right)\mathclose{}-g\mathopen{}\left( t\right)\mathclose{}\right\rvert\mathclose{}\lt ε
\)
| modulus s real number - minus t real number | modulus < less than δ positive real number \(
\mathopen{}\left\lvert{}s-t\right\rvert\mathclose{}\lt δ
\)
a real number = equals λ real number 0 zero < less than λ real number 1 one < less than ⋯ < less than λ real number n integer = equals b real number \(
a= {λ}_{0}\lt {λ}_{1}\lt \dotsb\lt {λ}_{n}= b
\)
λ real number j integer - minus λ real number j integer - minus 1 one < less than δ positive real number \(
{λ}_{j}-{λ}_{j-1}\lt δ
\) j integer \( j \)
a real number = equals μ real number 0 zero < less than μ real number 1 one < less than ⋯ < less than μ real number r real number = equals b real number \(
a= {μ}_{0}\lt {μ}_{1}\lt \dotsb\lt {μ}_{r}= b
\) λ real number \( λ \)
t real number j integer ∈ element of [ interval λ real number j integer - minus 1 one , λ real number j integer ] interval \(
{t}_{j}\in \mathopen{}\left[{λ}_{j-1}, {λ}_{j}\right]\mathclose{}
\) μ real number \( μ \)
s real number i integer ∈ element of [ interval μ real number i integer - minus 1 one , μ real number i integer ] interval \(
{s}_{i}\in \mathopen{}\left[{μ}_{i-1}, {μ}_{i}\right]\mathclose{}
\)
T operator = equals ∑ summation j integer = 1 one n integer
g continuous function ( t real number j integer ) times ( P projection λ real number j integer - minus P projection λ real number j integer - minus 1 one ) \(
T= \sum_{j=1}^{n}{}
g\mathopen{}\left( {t}_{j}\right)\mathclose{}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{}
\)
S operator = equals ∑ summation i integer = 1 one r real number
g continuous function ( s real number i integer ) times ( P projection μ real number i integer - minus P projection μ real number i integer - minus 1 one ) \(
S= \sum_{i=1}^{r}{}
g\mathopen{}\left( {s}_{i}\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{}
\)
‖ S operator - minus T operator ‖ < less than ε positive real number \(
\mathopen{}\left\lVert{}S-T\right\rVert\mathclose{}\lt ε
\)
‖ S operator - minus T operator ‖ = equals ‖ ∑ summation j integer = 1 one n integer
( S operator - minus T operator ) times ( P projection λ real number j integer - minus P projection λ real number j integer - minus 1 one ) ‖ = equals max maximum j integer
‖ ( S operator - minus T operator ) times ( P projection λ real number j integer - minus P projection λ real number j integer - minus 1 one ) ‖ \[
\mathopen{}\left\lVert{}S-T\right\rVert\mathclose{}= \mathopen{}\left\lVert{}\sum_{j=1}^{n}{}
\mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{}
\right\rVert\mathclose{}= \max_{j}{}
\mathopen{}\left\lVert{}\mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{λ}_{j}}-{P}_{{λ}_{j-1}}\right)\mathclose{}\right\rVert\mathclose{}
\] ε positive real number \( ε \)
j integer = equals 1 one \(
j= 1
\)
[ interval λ real number 0 zero , λ real number 1 one ] interval \(
\mathopen{}\left[{λ}_{0}, {λ}_{1}\right]\mathclose{}
\)
λ real number 0 zero = equals μ real number 0 zero < less than μ real number 1 one < less than ⋯ < less than μ real number k integer = equals λ real number 1 one \(
{λ}_{0}= {μ}_{0}\lt {μ}_{1}\lt \dotsb\lt {μ}_{k}= {λ}_{1}
\)
‖ ( S operator - minus T operator ) times ( P projection λ real number 1 one - minus P projection λ real number 0 zero ) ‖ = equals ‖ ∑ summation i integer = 1 one k integer
( S operator - minus T operator ) times ( P projection μ real number i integer - minus P projection μ real number i integer - minus 1 one ) ‖ = equals max maximum 1 one ≤ less than or equal to i integer ≤ less than or equal to k integer
‖ ( S operator - minus T operator ) times ( P projection μ real number i integer - minus P projection μ real number i integer - minus 1 one ) ‖ = equals max maximum i integer
‖ ( g continuous function ( s real number i integer ) - minus g continuous function ( t real number 1 one ) ) times ( P projection μ real number i integer - minus P projection μ real number i integer - minus 1 one ) ‖ ≤ less than or equal to max maximum i integer
| modulus g continuous function ( s real number i integer ) - minus g continuous function ( t real number 1 one ) | modulus < less than ε positive real number , \[
\mathopen{}\left\lVert{}\mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{λ}_{1}}-{P}_{{λ}_{0}}\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}\sum_{i=1}^{k}{}
\mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{}
\right\rVert\mathclose{}= \max_{1\leq i\leq k}{}
\mathopen{}\left\lVert{}\mathopen{}\left(S-T\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{}\right\rVert\mathclose{}
= \max_{i}{}
\mathopen{}\left\lVert{}\mathopen{}\left(g\mathopen{}\left( {s}_{i}\right)\mathclose{}-g\mathopen{}\left( {t}_{1}\right)\mathclose{}\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{}\right\rVert\mathclose{}
\leq \max_{i}{}
\mathopen{}\left\lvert{}g\mathopen{}\left( {s}_{i}\right)\mathclose{}-g\mathopen{}\left( {t}_{1}\right)\mathclose{}\right\rvert\mathclose{}
\lt ε
\text{,}
\]
| modulus s real number i integer - minus t real number 1 one | modulus < less than δ positive real number \(
\mathopen{}\left\lvert{}{s}_{i}-{t}_{1}\right\rvert\mathclose{}\lt δ
\) i integer \( i \)
Remark IV.7
Because
ℒ bounded linear operators ( H Hilbert space ) \(
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
T operator g continuous function ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
{T}_{g}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
T operator g continuous function = equals ∫ integral a real number b real number g continuous function ( λ real number ) d P projection λ real number \(
{T}_{g}= \int _{a}^{b}{}g\mathopen{}\left( λ\right)\mathclose{}\,\mathrm{d}{P}_{λ}
\)
ε positive real number > greater than 0 zero \(
ε\gt 0
\) F partition \( F \)
‖ T operator g continuous function - minus S operator ‖ < less than ε positive real number \(
\mathopen{}\left\lVert{}{T}_{g}-S\right\rVert\mathclose{}\lt ε
\)
S operator = equals ∑ summation
g continuous function ( s real number i integer ) times ( P projection μ real number i integer - minus P projection μ real number i integer - minus 1 one ) \(
S= \sum{}
g\mathopen{}\left( {s}_{i}\right)\mathclose{}\mathopen{}\left({P}_{{μ}_{i}}-{P}_{{μ}_{i-1}}\right)\mathclose{}
\)
{ set μ real number 0 zero μ real number 1 one … μ real number n integer } set \(
\mathopen{}\left\{\, {μ}_{0}, {μ}_{1}, \dotsc, {μ}_{n}\,\right\}\mathclose{}
\) F partition \( F \)
Some easy properties:
T operator g continuous function \(
{T}_{g}
\)
P projection λ real number \(
{P}_{λ}
\)
T operator c real number times g continuous function 1 one + plus g continuous function 2 two = equals c real number times T operator g continuous function 1 one + plus T operator g continuous function 2 two \(
{T}_{c{g}_{1}+{g}_{2}}= c{T}_{{g}_{1}}+{T}_{{g}_{2}}
\)
T operator g continuous function * = equals T operator g continuous function ¯ complex conjugate \(
{T}_{g}^{*}= {T}_{\overline{g}}
\)
T operator g continuous function 1 one times g continuous function 2 two = equals T operator g continuous function 1 one times T operator g continuous function 2 two \(
{T}_{{g}_{1}{g}_{2}}= {T}_{{g}_{1}}{T}_{{g}_{2}}
\)
‖ T operator ( g continuous function ) ‖ ≤ less than or equal to max maximum [ interval a real number , b real number ] interval | modulus g continuous 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
λ real number 0 zero < less than λ real number 1 one < less than ⋯ < less than λ real number n integer \(
{λ}_{0}\lt {λ}_{1}\lt \dotsb\lt {λ}_{n}
\)
g continuous function 1 one \(
{g}_{1}
\)
g continuous function 2 two \(
{g}_{2}
\)
∑ summation
g continuous function 1 ( t real number j integer ) Δ difference P projection λ real number j integer
\(
\sum{}
{g}_{1}\mathopen{}\left( {t}_{j}\right)\mathclose{}
\, \Delta
{P}_{{λ}_{j}}
\)
∑ summation
g continuous function 2 ( t real number j integer ) Δ difference P projection λ real number j integer
\(
\sum{}
{g}_{2}\mathopen{}\left( {t}_{j}\right)\mathclose{}
\, \Delta
{P}_{{λ}_{j}}
\)
T operator g continuous function 1 one \(
{T}_{{g}_{1}}
\)
T operator g continuous function 2 two \(
{T}_{{g}_{2}}
\)
Δ difference P projection λ real number i integer = equals P projection λ real number i integer - minus P projection λ real number i integer - minus 1 one \(
\Delta {P}_{{λ}_{i}}= {P}_{{λ}_{i}}-{P}_{{λ}_{i-1}}
\)
( ∑ summation
g continuous function 1 ( t real number j integer ) Δ difference P projection λ real number j integer
) times ( ∑ summation
g continuous function 2 ( t real number j integer ) Δ difference P projection λ real number j integer
) = equals ∑ summation
g continuous function 1 ( t real number j integer ) g continuous function 2 ( t real number j integer ) Δ difference P projection λ real number j integer
. \[
\mathopen{}\left(\sum{}
{g}_{1}\mathopen{}\left( {t}_{j}\right)\mathclose{}
\, \Delta
{P}_{{λ}_{j}}
\right)\mathclose{}\mathopen{}\left(\sum{}
{g}_{2}\mathopen{}\left( {t}_{j}\right)\mathclose{}
\, \Delta
{P}_{{λ}_{j}}
\right)\mathclose{}= \sum{}
{g}_{1}\mathopen{}\left( {t}_{j}\right)\mathclose{}{g}_{2}\mathopen{}\left( {t}_{j}\right)\mathclose{}
\, \Delta
{P}_{{λ}_{j}}
\text{.}
\]
P projection λ real number j integer times P projection λ real number k integer = equals 0 zero \(
{P}_{{λ}_{j}}
{P}_{{λ}_{k}}
= 0
\)
j integer ≠ not equal to k integer \(
j\neq k
\)
Theorem IV.8
Let
A self-adjoint operator = equals A self-adjoint operator * ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
A= A^{*}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
σ ( A self-adjoint operator ) ⊆ subset ( interval a real number , b real number ) interval \(
σ\mathopen{}\left( A\right)\mathclose{}\subseteq \mathopen{}\left(a, b\right)\mathclose{}
\)
λ real number ↦ is mapped to P projection λ real number \(
λ\mapsto {P}_{λ}
\) Theorem IV.5 . Then for
g continuous function ∈ element of C space of continuous functions ( [ interval a real number , b real number ] interval ) \(
g\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{}
\)
∫ integral a real number b real number
g continuous function ( λ real number ) d P projection λ real number = equals ( g continuous function | restricted to σ ( A self-adjoint operator ) ) ( A self-adjoint operator ) \(
\int _{a}^{b}{}
g\mathopen{}\left( λ\right)\mathclose{}
\,\mathrm{d}{P}_{λ}= \mathopen{}\left(g|σ\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( A\right)\mathclose{}
\) functional calculus on
C space of continuous functions ( σ ( A self-adjoint operator ) ) \(
\mathrm{C}\mathopen{}\left( σ\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
We have shown that
g continuous function ↦ is mapped to ∫ integral a real number b real number
g continuous function ( λ real number ) d P projection λ real number ≡ equivalent T operator g continuous function \(
g\mapsto \int _{a}^{b}{}
g\mathopen{}\left( λ\right)\mathclose{}
\,\mathrm{d}{P}_{λ}\equiv {T}_{g}
\)
C space of continuous functions ( [ interval a real number , b real number ] interval ) \(
\mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{}
\)
ℒ bounded linear operators ( H Hilbert space ) \(
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
‖ T operator g continuous function ‖ ≤ less than or equal to max maximum [ interval a real number , b real number ] interval | modulus g continuous function | modulus \(
\mathopen{}\left\lVert{}{T}_{g}\right\rVert\mathclose{}\leq \max_{\mathopen{}\left[a, b\right]\mathclose{}}{}\mathopen{}\left\lvert{}g\right\rvert\mathclose{}
\)
∫ integral a real number b real number
1 one d P projection λ real number = equals I \(
\int _{a}^{b}{}
1
\,\mathrm{d}{P}_{λ}= I
\)
∫ integral a real number b real number
λ real number d P projection λ real number = equals A self-adjoint operator \(
\int _{a}^{b}{}
λ
\,\mathrm{d}{P}_{λ}= A
\)
p polynomial ( x vector ) \(
p\mathopen{}\left( x\right)\mathclose{}
\)
∫ integral a real number b real number
p polynomial ( A self-adjoint operator ) d P projection λ real number = equals p polynomial ( A self-adjoint operator ) \(
\int _{a}^{b}{}
p\mathopen{}\left( A\right)\mathclose{}
\,\mathrm{d}{P}_{λ}= p\mathopen{}\left( A\right)\mathclose{}
\)
g continuous function ∈ element of C space of continuous functions ( [ interval a real number , b real number ] interval ) \(
g\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{}
\)
ĝ = equals g continuous function | restricted to σ ( A self-adjoint operator ) \(
ĝ= g|σ\mathopen{}\left( A\right)\mathclose{}
\)
( sequence
p polynomial n integer ) sequence \(
\mathopen{}\left(
{p}_{n}
\right)\mathclose{}
\)
p polynomial n integer → converges to g continuous function \(
{p}_{n} \to g
\)
[ interval a real number , b real number ] interval \(
\mathopen{}\left[a, b\right]\mathclose{}
\)
p polynomial n integer ( A self-adjoint operator ) = equals ∫ integral a real number b real number
p polynomial n integer ( λ real number ) d P projection λ real number → converges to ∫ integral a real number b real number
g continuous function ( λ real number ) d P projection λ real number \(
{p}_{n}\mathopen{}\left( A\right)\mathclose{}= \int _{a}^{b}{}
{p}_{n}\mathopen{}\left( λ\right)\mathclose{}
\,\mathrm{d}{P}_{λ} \to \int _{a}^{b}{}
g\mathopen{}\left( λ\right)\mathclose{}
\,\mathrm{d}{P}_{λ}
\)
p polynomial n integer → converges to ĝ \(
{p}_{n} \to ĝ
\)
σ ( A self-adjoint operator ) \(
σ\mathopen{}\left( A\right)\mathclose{}
\)
p polynomial n integer ( A self-adjoint operator ) → converges to ĝ ( A self-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
g continuous function ∈ element of C space of continuous functions ( [ interval a real number , b real number ] interval ) \(
g\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[a, b\right]\mathclose{}\right)\mathclose{}
\)
∫ integral a real number b real number
g continuous function ( λ real number ) d P projection λ real number \(
\int _{a}^{b}{}
g\mathopen{}\left( λ\right)\mathclose{}
\,\mathrm{d}{P}_{λ}
\) g continuous function \( g \)
σ ( A self-adjoint operator ) \(
σ\mathopen{}\left( A\right)\mathclose{}
\)
σ ( A self-adjoint operator ) \(
σ\mathopen{}\left( A\right)\mathclose{}
\)
λ real number ↦ is mapped to P projection λ real number \(
λ\mapsto {P}_{λ}
\)
Eigenvalues of A self-adjoint operator \( A \)
Lemma IV.10
Let
( sequence E projection n integer ) sequence \(
\mathopen{}\left({E}_{n}\right)\mathclose{}
\)
E projection n integer ≥ greater than or equal to E projection n integer + plus 1 one \(
{E}_{n}\geq {E}_{n+1}
\) n integer \( n \) E projection \( E \)
⋂ intersection n integer E projection n integer ( H Hilbert space ) \(
\bigcap_{n}{}{E}_{n}\mathopen{}\left( H\right)\mathclose{}
\)
E projection n integer ( ξ vector ) → converges to E projection ( ξ vector ) \(
{E}_{n}\mathopen{}\left( ξ\right)\mathclose{} \to E\mathopen{}\left( ξ\right)\mathclose{}
\)
ξ vector ∈ element of H Hilbert space \(
ξ\in H
\)
The nonnegative sequence
‖ ( E projection n integer - minus E projection ) ( ξ vector ) ‖ \(
\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}
\) r real number \( r \)
ε positive real number > greater than 0 zero \(
ε\gt 0
\) N integer \( N \)
‖ ( E projection n integer - minus E projection ) ( ξ vector ) ‖ 2 two < less than r real number 2 two + plus ε positive real number \(
{\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}\lt {r}^{2}+ε
\)
n integer ≥ greater than or equal to N integer \(
n\geq N
\)
m integer > greater than n integer ≥ greater than or equal to N integer \(
m\gt n\geq N
\)
( E projection n integer - minus E projection m integer ) ( ξ vector ) \(
\mathopen{}\left({E}_{n}-{E}_{m}\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}
\)
( E projection m integer - minus E projection ) ( ξ vector ) \(
\mathopen{}\left({E}_{m}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}
\)
‖ ( E projection n integer - minus E projection ) ( ξ vector ) ‖ 2 two = equals
‖ ( E projection n integer - minus E projection m integer ) ( ξ vector ) ‖ 2 two + plus
‖ ( E projection m integer - minus E projection ) ( ξ vector ) ‖ 2 two \[
{\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-{E}_{m}\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}+{\mathopen{}\left\lVert{}\mathopen{}\left({E}_{m}-E\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}
\]
‖ ( E projection n integer - minus E projection m integer ) ( ξ vector ) ‖ 2 two \(
{\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-{E}_{m}\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}
\)
[ interval r real number 2 two , r real number 2 two + plus ε positive real number ) interval \(
\mathopen{}\left[{r}^{2}, {r}^{2}+ε\right)\mathclose{}
\)
‖ ( E projection n integer - minus E projection m integer ) ( ξ vector ) ‖ 2 two < less than ε positive real number \(
{\mathopen{}\left\lVert{}\mathopen{}\left({E}_{n}-{E}_{m}\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}\lt ε
\)
( sequence
E projection n integer ( ξ vector ) ) sequence \(
\mathopen{}\left(
{E}_{n}\mathopen{}\left( ξ\right)\mathclose{}
\right)\mathclose{}
\) η vector \( η \) k integer \( k \)
E projection k integer ( η vector ) = equals lim limit n integer
E projection k integer ( E projection n integer ( ξ vector ) ) = equals lim limit n integer
E projection n integer ( ξ vector ) = equals η vector \(
{E}_{k}\mathopen{}\left( η\right)\mathclose{}= \lim_{n}{}
{E}_{k}\mathopen{}\left( {E}_{n}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}
= \lim_{n}{}
{E}_{n}\mathopen{}\left( ξ\right)\mathclose{}
= η
\)
η vector = equals E projection ( η vector ) = equals lim limit n integer
E projection ( E projection n integer ( ξ vector ) ) = equals E projection ( ξ vector ) \(
η= E\mathopen{}\left( η\right)\mathclose{}= \lim_{n}{}
E\mathopen{}\left( {E}_{n}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}
= E\mathopen{}\left( ξ\right)\mathclose{}
\)
Remark IV.11
For
λ real number ∈ element of R real numbers \(
λ\in \mathbb{R}
\)
Q projection λ real number \(
{Q}_{λ}
\)
⋃ union θ real number < less than λ real number
P projection θ real number ( H Hilbert space ) ¯ complex conjugate \(
\overline{
\bigcup_{θ\lt λ}{}
{P}_{θ}\mathopen{}\left( H\right)\mathclose{}
}
\)
R projection λ real number \(
{R}_{λ}
\)
⋂ intersection θ real number > greater than λ real number
P projection θ real number ( H Hilbert space ) \(
\bigcap_{θ\gt λ}{}
{P}_{θ}\mathopen{}\left( H\right)\mathclose{}
\)
Q projection λ real number ≤ less than or equal to P projection λ real number ≤ less than or equal to R projection λ real number \(
{Q}_{λ}\leq {P}_{λ}\leq {R}_{λ}
\)
lim limit θ real number → λ real number +
P projection θ real number ( ξ vector ) = equals R projection λ real number ( ξ vector ) \(
\lim_{θ\to{λ}^{\mathrm{+}}}{}
{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}
= {R}_{λ}\mathopen{}\left( ξ\right)\mathclose{}
\) ξ vector \( ξ \)
Proposition IV.12
For all λ real number \( λ \)
R projection λ real number - minus Q projection λ real number \(
{R}_{λ}-{Q}_{λ}
\)
Ker kernel ( A self-adjoint operator - minus λ real number ) \(
\operatorname{Ker}\mathopen{}\left( A-λ\right)\mathclose{}
\) λ real number \( λ \) A self-adjoint operator \( A \)
Q projection λ real number ≠ not equal to R projection λ real number \(
{Q}_{λ}\neq {R}_{λ}
\)
For any ξ vector \( ξ \)
〈 〈 ( A self-adjoint operator - minus λ real number ) 2 two ( ξ vector ) 〉 , ξ vector 〉 = equals ∫ integral a real number b real number
( θ real number - minus λ real number ) 2 two d
〈 P projection θ real number ( ξ vector ) , ξ vector 〉 \(
\mathopen{}\left\langle{}\mathopen{}\left\langle{}{\mathopen{}\left(A-λ\right)\mathclose{}}^{2}\mathopen{}\left( ξ\right)\mathclose{}\right\rangle\mathclose{}, ξ\right\rangle\mathclose{}= \int _{a}^{b}{}
{\mathopen{}\left(θ-λ\right)\mathclose{}}^{2}
\,\mathrm{d}
\mathopen{}\left\langle{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}, ξ\right\rangle\mathclose{}
\)
〈 P projection θ real number ( ξ vector ) , ξ vector 〉 \(
\mathopen{}\left\langle{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}, ξ\right\rangle\mathclose{}
\)
‖ P projection θ real number ( ξ vector ) ‖ 2 two \(
{\mathopen{}\left\lVert{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}
\)
θ real number > greater than λ real number \(
θ\gt λ
\)
θ real number < less than λ real number \(
θ\lt λ
\)
‖ P projection θ real number ( ξ vector ) ‖ 2 two = equals ‖ ξ vector ‖ 2 two \(
{\mathopen{}\left\lVert{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}\right\rVert\mathclose{}}^{2}= {\mathopen{}\left\lVert{}ξ\right\rVert\mathclose{}}^{2}
\)
P projection θ real number ( ξ vector ) = equals ξ vector \(
{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}= ξ
\)
θ real number > greater than λ real number \(
θ\gt λ
\)
P projection θ real number ( ξ vector ) = equals 0 zero \(
{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}= 0
\)
θ real number < less than λ real number \(
θ\lt λ
\)
R projection λ real number ( ξ vector ) = equals ξ vector \(
{R}_{λ}\mathopen{}\left( ξ\right)\mathclose{}= ξ
\)
Q projection λ real number ( ξ vector ) = equals 0 zero \(
{Q}_{λ}\mathopen{}\left( ξ\right)\mathclose{}= 0
\)
Ker kernel ( A self-adjoint operator - minus λ real number ) ⊆ subset ( R projection λ real number - minus Q projection λ real number ) ( H Hilbert space ) \(
\operatorname{Ker}\mathopen{}\left( A-λ\right)\mathclose{}\subseteq \mathopen{}\left({R}_{λ}-{Q}_{λ}\right)\mathclose{}\mathopen{}\left( H\right)\mathclose{}
\)
R projection λ real number ( ξ vector ) = equals ξ vector \(
{R}_{λ}\mathopen{}\left( ξ\right)\mathclose{}= ξ
\)
Q projection λ real number ( ξ vector ) = equals 0 zero \(
{Q}_{λ}\mathopen{}\left( ξ\right)\mathclose{}= 0
\)
θ real number ↦ is mapped to 〈 P projection θ real number ( ξ vector ) , ξ vector 〉 \(
θ\mapsto \mathopen{}\left\langle{}{P}_{θ}\mathopen{}\left( ξ\right)\mathclose{}, ξ\right\rangle\mathclose{}
\) 0 zero \( 0 \)
‖ ξ vector ‖ 2 two \(
{\mathopen{}\left\lVert{}ξ\right\rVert\mathclose{}}^{2}
\) λ real number \( λ \)
‖ ξ vector ‖ 2 two \(
{\mathopen{}\left\lVert{}ξ\right\rVert\mathclose{}}^{2}
\) λ real number \( λ \) 0 zero \( 0 \)
( A self-adjoint operator - minus λ real number ) ( ξ vector ) = equals 0 zero \(
\mathopen{}\left(A-λ\right)\mathclose{}\mathopen{}\left( ξ\right)\mathclose{}= 0
\)
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 T operator \( T \)
{ set ξ function ∈ element of L 2 Lebesgue space ( R real numbers ) | such that
∫ integral R real numbers
x vector 2 two times
| modulus ξ function ( x vector ) | modulus 2 two d x vector < less than ∞ infinity } set \(
\mathopen{}\left\{\, ξ\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}\right)\mathclose{}\,\middle\vert\,
, \int _{\mathbb{R}}{}
{x}^{2}{\mathopen{}\left\lvert{}ξ\mathopen{}\left( x\right)\mathclose{}\right\rvert\mathclose{}}^{2}
\,\mathrm{d}x\lt \infty,
\,\right\}\mathclose{}
\)
( T operator ( ξ function ) ) ( x vector ) = equals x vector times ξ function ( x vector ) \(
\mathopen{}\left(T\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= xξ\mathopen{}\left( x\right)\mathclose{}
\)