Lecture Notes in Functional Analysis
by William L. Paschke
edition 0.9
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F. Self-Adjoint Operators
Call an operator
A self-adjoint operator ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
A\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\) self-adjoint if
A self-adjoint operator = equals A self-adjoint operator * \(
A= A^{*}
\)
T linear map * times T linear map \(
T^{*}T
\)
T linear map + plus T linear map * \(
T+ T^{*}
\)
i imaginary unit times ( T linear map - minus T linear map * ) \(
\mathrm{i}\mathopen{}\left(T- T^{*}\right)\mathclose{}
\)
T linear map ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
T\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\) polynomial calculus for a self-adjoint operator extends to a functional calculus for arbitrary complex continuous functions on the spectrum of the operator. For the real spectrum result, we will need the following lemma.
Lemma II.53 Let
T linear map ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
T\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
r real number > greater than 0 zero \(
r\gt 0
\)
| modulus 〈 T linear map ( x vector ) , x vector 〉 | modulus ≥ greater than or equal to r real number times ‖ x vector ‖ 2 two \(
\mathopen{}\left\lvert{}\mathopen{}\left\langle{}T\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}\right\rvert\mathclose{}\geq r{\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}
\)
x vector ∈ element of H Hilbert space \(
x\in H
\) T linear map \( T \)
By Cauchy-Schwartz,
‖ T linear map ( x vector ) ‖ times ‖ x vector ‖ ≥ greater than or equal to r real number times ‖ x vector ‖ 2 two \(
\mathopen{}\left\lVert{}T\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}\mathopen{}\left\lVert{}x\right\rVert\mathclose{}\geq r{\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}
\)
‖ T linear map ( x vector ) ‖ ≥ greater than or equal to r real number times ‖ x vector ‖ \(
\mathopen{}\left\lVert{}T\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}\geq r\mathopen{}\left\lVert{}x\right\rVert\mathclose{}
\) x vector \( x \) T linear map \( T \)
T linear map ( x vector n integer ) → converges to y vector \(
T\mathopen{}\left( {x}_{n}\right)\mathclose{} \to y
\)
( sequence x vector n integer ) sequence \(
\mathopen{}\left({x}_{n}\right)\mathclose{}
\)
‖ x vector n integer - minus x vector m integer ‖ ≤ less than or equal to 1 one r real number times ‖ T linear map ( x vector n integer ) - minus T linear map ( x vector m integer ) ‖ \(
\mathopen{}\left\lVert{}{x}_{n}-{x}_{m}\right\rVert\mathclose{}\leq \frac{1}{r}\mathopen{}\left\lVert{}T\mathopen{}\left( {x}_{n}\right)\mathclose{}-T\mathopen{}\left( {x}_{m}\right)\mathclose{}\right\rVert\mathclose{}
\)
x vector n integer → converges to x vector \(
{x}_{n} \to x
\)
y vector = equals T linear map ( x vector ) \(
y= T\mathopen{}\left( x\right)\mathclose{}
\) T linear map \( T \)
w vector ∈ element of T linear map ( H Hilbert space ) ⊥ \(
w\in {T\mathopen{}\left( H\right)\mathclose{}}^{\perp}
\)
〈 w vector , T linear map ( x vector ) 〉 = equals 0 zero \(
\mathopen{}\left\langle{}w, T\mathopen{}\left( x\right)\mathclose{}\right\rangle\mathclose{}= 0
\) x vector \( x \)
〈 T linear map * ( w vector ) , x vector 〉 = equals 0 zero \(
\mathopen{}\left\langle{} T^{*}\mathopen{}\left( w\right)\mathclose{}, x\right\rangle\mathclose{}= 0
\)
T linear map * ( w vector ) = equals 0 zero \(
T^{*}\mathopen{}\left( w\right)\mathclose{}= 0
\)
r real number times ‖ w vector ‖ 2 two ≤ less than or equal to | modulus 〈 T linear map ( w vector ) , w vector 〉 | modulus = equals | modulus 〈 w vector , T linear map * ( w vector ) 〉 | modulus = equals 0 zero \(
r{\mathopen{}\left\lVert{}w\right\rVert\mathclose{}}^{2}\leq \mathopen{}\left\lvert{}\mathopen{}\left\langle{}T\mathopen{}\left( w\right)\mathclose{}, w\right\rangle\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\mathopen{}\left\langle{}w, T^{*}\mathopen{}\left( w\right)\mathclose{}\right\rangle\mathclose{}\right\rvert\mathclose{}= 0
\)
T linear map ( H Hilbert space ) = equals H Hilbert space \(
T\mathopen{}\left( H\right)\mathclose{}= H
\)
T linear map ( x vector ) = equals 0 zero \(
T\mathopen{}\left( x\right)\mathclose{}= 0
\)
r real number times ‖ x vector ‖ 2 two = equals 0 zero \(
r{\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}= 0
\)
x vector = equals 0 zero \(
x= 0
\) T linear map \( T \)
T linear map − 1 inverse : maps H Hilbert space → to H Hilbert space \(
{T}^{-1} : H \to H
\)
r real number times ‖ T linear map − 1 inverse ( x vector ) ‖ ≤ less than or equal to ‖ T linear map ( T linear map − 1 inverse ( x vector ) ) ‖ = equals ‖ x vector ‖ \(
r\mathopen{}\left\lVert{}{T}^{-1}\mathopen{}\left( x\right)\mathclose{}\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}T\mathopen{}\left( {T}^{-1}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}\right\rVert\mathclose{}= \mathopen{}\left\lVert{}x\right\rVert\mathclose{}
\)
T linear map − 1 inverse ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
{T}^{-1}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
‖ T linear map − 1 inverse ‖ ≤ less than or equal to 1 one r real number \(
\mathopen{}\left\lVert{}{T}^{-1}\right\rVert\mathclose{}\leq \frac{1}{r}
\)
Proposition II.54
An operator
A self-adjoint operator ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
A\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
〈 A self-adjoint operator ( x vector ) , x vector 〉 \(
\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}
\)
x vector ∈ element of H Hilbert space \(
x\in H
\)
(⇒)
〈 A self-adjoint operator ( x vector ) , x vector 〉 = equals 〈 x vector , A self-adjoint operator ( x vector ) 〉 \(
\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, A\mathopen{}\left( x\right)\mathclose{}\right\rangle\mathclose{}
\)
〈 A self-adjoint operator ( x vector ) , x vector 〉 = equals 〈 A self-adjoint operator ( x vector ) , x vector 〉 ¯ complex conjugate \(
\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}= \overline{\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}}
\)
(⇐)
〈 A self-adjoint operator ( x vector + plus y vector ) , x vector + plus y vector 〉 = equals 〈 A self-adjoint operator ( x vector ) , x vector 〉 + plus 〈 A self-adjoint operator ( x vector ) , y vector 〉 + plus 〈 A self-adjoint operator ( y vector ) , x vector 〉 + plus 〈 A self-adjoint operator ( y vector ) , y vector 〉 ∈ element of R real numbers \(
\mathopen{}\left\langle{}A\mathopen{}\left( x+y\right)\mathclose{}, x+y\right\rangle\mathclose{}= \mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}+\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}+\mathopen{}\left\langle{}A\mathopen{}\left( y\right)\mathclose{}, x\right\rangle\mathclose{}+\mathopen{}\left\langle{}A\mathopen{}\left( y\right)\mathclose{}, y\right\rangle\mathclose{}\in \mathbb{R}
\)
〈 A self-adjoint operator ( x vector ) , y vector 〉 + plus 〈 A self-adjoint operator ( y vector ) , x vector 〉 ∈ element of R real numbers \(
\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}+\mathopen{}\left\langle{}A\mathopen{}\left( y\right)\mathclose{}, x\right\rangle\mathclose{}\in \mathbb{R}
\)
Im imaginary part ( 〈 A self-adjoint operator ( x vector ) , y vector 〉 ) = equals −
Im imaginary part ( 〈 A self-adjoint operator ( y vector ) , x vector 〉 ) \(
\Im\mathopen{}\left( \mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}\right)\mathclose{}= {-}
\Im\mathopen{}\left( \mathopen{}\left\langle{}A\mathopen{}\left( y\right)\mathclose{}, x\right\rangle\mathclose{}\right)\mathclose{}
\) y vector \( y \)
i imaginary unit times y vector \(
\mathrm{i}y
\)
− i imaginary unit times 〈 A self-adjoint operator ( x vector ) , y vector 〉 + plus i imaginary unit times 〈 A self-adjoint operator ( y vector ) , x vector 〉 ∈ element of R real numbers \(
{-}\mathrm{i}\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}+\mathrm{i}\mathopen{}\left\langle{}A\mathopen{}\left( y\right)\mathclose{}, x\right\rangle\mathclose{}\in \mathbb{R}
\)
Re real part ( 〈 A self-adjoint operator ( x vector ) , y vector 〉 ) = equals Re real part ( 〈 A self-adjoint operator ( y vector ) , x vector 〉 ) \(
\Re\mathopen{}\left( \mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}\right)\mathclose{}= \Re\mathopen{}\left( \mathopen{}\left\langle{}A\mathopen{}\left( y\right)\mathclose{}, x\right\rangle\mathclose{}\right)\mathclose{}
\)
〈 A self-adjoint operator ( y vector ) , x vector 〉 ¯ complex conjugate = equals 〈 A self-adjoint operator ( x vector ) , y vector 〉 \(
\overline{
\mathopen{}\left\langle{}A\mathopen{}\left( y\right)\mathclose{}, x\right\rangle\mathclose{}
}= \mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}
\)
〈 A self-adjoint operator ( x vector ) , y vector 〉 = equals 〈 x vector , A self-adjoint operator ( y vector ) 〉 = equals 〈 A self-adjoint operator * ( x vector ) , y vector 〉 \(
\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, A\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}= \mathopen{}\left\langle{} A^{*}\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}
\)
〈 ( A self-adjoint operator - minus A self-adjoint operator * ) ( x vector ) , y vector 〉 = equals 0 zero \(
\mathopen{}\left\langle{}\mathopen{}\left(A- A^{*}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}= 0
\)
Proposition II.55
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 R real numbers \(
\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\subseteq \mathbb{R}
\)
Let
λ complex number = equals α complex number + plus i imaginary unit times β complex number \(
λ= α+\mathrm{i}β
\) α complex number \( α \) β complex number \( β \) R real numbers \( \mathbb{R} \)
β complex number ≠ not equal to 0 zero \(
β\neq 0
\)
| modulus 〈 ( λ complex number - minus A self-adjoint operator ) ( x vector ) , x vector 〉 | modulus = equals | modulus 〈 ( α complex number - minus A self-adjoint operator ) ( x vector ) , x vector 〉 + plus i imaginary unit times β complex number times 〈 x vector , x vector 〉 | modulus ≥ greater than or equal to | modulus β complex number | modulus times 〈 x vector , x vector 〉 = equals | modulus β complex number | modulus times ‖ x vector ‖ 2 two \(
\mathopen{}\left\lvert{}\mathopen{}\left\langle{}\mathopen{}\left(λ-A\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\mathopen{}\left\langle{}\mathopen{}\left(α-A\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}, x\right\rangle\mathclose{}+\mathrm{i}β\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}\right\rvert\mathclose{}\geq \mathopen{}\left\lvert{}β\right\rvert\mathclose{}\mathopen{}\left\langle{}x, x\right\rangle\mathclose{}= \mathopen{}\left\lvert{}β\right\rvert\mathclose{}{\mathopen{}\left\lVert{}x\right\rVert\mathclose{}}^{2}
\)
λ complex number - minus A self-adjoint operator ∈ element of
ℒ bounded linear operators ( X normed linear space ) − 1 invertible elements \(
λ-A\in {
\mathcal{L}\mathopen{}\left( X\right)\mathclose{}
}^{-1}
\) Lemma II.53 .
Proposition II.56
Let T linear map \( T \) p polynomial with complex coefficients \( p \)
p polynomial with complex coefficients ( T linear map ) \(
p\mathopen{}\left( T\right)\mathclose{}
\)
‖ p polynomial with complex coefficients ( T linear map ) ‖ = equals max maximum λ complex number ∈ element of σ ( T linear map ) | modulus p polynomial with complex coefficients ( λ complex number ) | modulus \(
\mathopen{}\left\lVert{}p\mathopen{}\left( T\right)\mathclose{}\right\rVert\mathclose{}= \max_{λ\in \mathop{\sigma}\mathopen{}\left( T\right)\mathclose{}}{}\mathopen{}\left\lvert{}p\mathopen{}\left( λ\right)\mathclose{}\right\rvert\mathclose{}
\)
Notice that
p polynomial with complex coefficients ( T linear map ) * = equals p polynomial with complex coefficients ¯ complex conjugate ( T linear map * ) \(
p\mathopen{}\left( T\right)\mathclose{}^{*}= \overline{p}\mathopen{}\left( T^{*}\right)\mathclose{}
\) p polynomial with complex coefficients ¯ complex conjugate \( \overline{p} \) p polynomial with complex coefficients \( p \) T linear map \( T \)
T linear map * \(
T^{*}
\)
T linear map j integer \(
{T}^{j}
\)
( T linear map * ) k integer \(
{\mathopen{}\left( T^{*}\right)\mathclose{}}^{k}
\) j integer \( j \) k integer \( k \) 0 zero \( 0 \)
p polynomial with complex coefficients ( T linear map ) \(
p\mathopen{}\left( T\right)\mathclose{}
\)
‖ p polynomial with complex coefficients ( T linear map ) ‖ \(
\mathopen{}\left\lVert{}p\mathopen{}\left( T\right)\mathclose{}\right\rVert\mathclose{}
\) Proposition II.44 and Proposition II.52 .
Now for the functional calculus for self-adjoint operators
A self-adjoint operator = equals A self-adjoint operator * ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
A= A^{*}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\) P polynomial subalgebra \( P \)
C space of continuous functions ( σ ( A self-adjoint operator ) ) \(
\mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\) P polynomial subalgebra \( P \)
σ ( A self-adjoint operator ) \(
\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}
\)
σ ( A self-adjoint operator ) \(
\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}
\) P polynomial subalgebra \( P \)
C space of continuous functions ( σ ( A self-adjoint operator ) ) \(
\mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\) Proposition II.56 , the map
p polynomial with complex coefficients ↦ is mapped to p polynomial with complex coefficients ( A self-adjoint operator ) \(
p\mapsto p\mathopen{}\left( A\right)\mathclose{}
\) P polynomial subalgebra \( P \)
ℒ bounded linear operators ( H Hilbert space ) \(
\mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
p polynomial with complex coefficients ( A self-adjoint operator ) * = equals p polynomial with complex coefficients ¯ complex conjugate ( A self-adjoint operator ) \(
p\mathopen{}\left( A\right)\mathclose{}^{*}= \overline{p}\mathopen{}\left( A\right)\mathclose{}
\)
( p polynomial with complex coefficients times q polynomial with complex coefficients ) ( A self-adjoint operator ) = equals p polynomial with complex coefficients ( A self-adjoint operator ) times q polynomial with complex coefficients ( A self-adjoint operator ) \(
\mathopen{}\left(pq\right)\mathclose{}\mathopen{}\left( A\right)\mathclose{}= p\mathopen{}\left( A\right)\mathclose{}q\mathopen{}\left( A\right)\mathclose{}
\)
Theorem II.57 Let
A self-adjoint operator = equals A self-adjoint operator * \(
A= A^{*}
\)
Φ isometric *-isomorphism : maps C space of continuous functions ( σ ( A self-adjoint operator ) ) → to ℒ bounded linear operators ( H Hilbert space ) \(
Φ : \mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{} \to \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
Φ isometric *-isomorphism ( p polynomial with complex coefficients ) = equals p polynomial with complex coefficients ( A self-adjoint operator ) \(
Φ\mathopen{}\left( p\right)\mathclose{}= p\mathopen{}\left( A\right)\mathclose{}
\)
p polynomial with complex coefficients ∈ element of P \(
p\in P
\)
As with any isometry mapping a dense subset of a metric space into a complete metric space, we obtain Φ isometric *-isomorphism \( Φ \)
Φ isometric *-isomorphism ( f function ) = equals lim limit n integer p polynomial with complex coefficients n integer ( A self-adjoint operator ) \(
Φ\mathopen{}\left( f\right)\mathclose{}= \lim_{n}{}{p}_{n}\mathopen{}\left( A\right)\mathclose{}
\)
( sequence p polynomial with complex coefficients n integer ) sequence \(
\mathopen{}\left({p}_{n}\right)\mathclose{}
\) f function \( f \)
σ ( A self-adjoint operator ) \(
\mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}
\) Φ isometric *-isomorphism \( Φ \)
Φ isometric *-isomorphism ( f function ) * = equals Φ isometric *-isomorphism ( f function ¯ complex conjugate ) \(
Φ\mathopen{}\left( f\right)\mathclose{}^{*}= Φ\mathopen{}\left( \overline{f}\right)\mathclose{}
\) f function \( f \)
C space of continuous functions ( σ ( A self-adjoint operator ) ) \(
\mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
A self-adjoint operator = equals A self-adjoint operator * \(
A= A^{*}
\)
p polynomial with complex coefficients ( A self-adjoint operator ) * = equals p polynomial with complex coefficients ¯ complex conjugate ( A self-adjoint operator ) \(
p\mathopen{}\left( A\right)\mathclose{}^{*}= \overline{p}\mathopen{}\left( A\right)\mathclose{}
\) p polynomial with complex coefficients \( p \)
Better notation: Write
f function ( A self-adjoint operator ) = equals Φ function ( f function ) \(
f\mathopen{}\left( A\right)\mathclose{}= Φ\mathopen{}\left( f\right)\mathclose{}
\)
f function ∈ element of C space of continuous functions ( σ ( A self-adjoint operator ) ) \(
f\in \mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}
\)
Lemma II.58
For
A self-adjoint operator = equals A self-adjoint operator * ∈ element of ℒ bounded linear operators ( H Hilbert space ) \(
A= A^{*}\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}
\)
λ complex number 1 one ≠ not equal to λ complex number 2 two ∈ element of R real numbers \(
{λ}_{1}\neq {λ}_{2}\in \mathbb{R}
\)
Ker kernel ( λ complex number 1 one - minus A self-adjoint operator ) ⊥ Ker kernel ( λ complex number 2 two - minus A self-adjoint operator ) \(
\operatorname{Ker}\mathopen{}\left( {λ}_{1}-A\right)\mathclose{}\perp \operatorname{Ker}\mathopen{}\left( {λ}_{2}-A\right)\mathclose{}
\)
A self-adjoint operator ( x vector i integer ) = equals λ complex number i integer times x vector i integer \(
A\mathopen{}\left( {x}_{i}\right)\mathclose{}= {λ}_{i}{x}_{i}
\)
λ complex number 1 one times 〈 x vector 1 one , x vector 2 two 〉 = equals 〈 A self-adjoint operator ( x vector 1 one ) , x vector 2 two 〉 = equals 〈 x vector 1 one , A self-adjoint operator ( x vector 2 two ) 〉 = equals λ complex number 2 two times 〈 x vector 1 one , x vector 2 two 〉 \[
{λ}_{1}\mathopen{}\left\langle{}{x}_{1}, {x}_{2}\right\rangle\mathclose{}= \mathopen{}\left\langle{}A\mathopen{}\left( {x}_{1}\right)\mathclose{}, {x}_{2}\right\rangle\mathclose{}= \mathopen{}\left\langle{}{x}_{1}, A\mathopen{}\left( {x}_{2}\right)\mathclose{}\right\rangle\mathclose{}= {λ}_{2}\mathopen{}\left\langle{}{x}_{1}, {x}_{2}\right\rangle\mathclose{}
\]
〈 x vector 1 one , x vector 2 two 〉 = equals 0 zero \(
\mathopen{}\left\langle{}{x}_{1}, {x}_{2}\right\rangle\mathclose{}= 0
\)
Lemma II.59
Let M closed invariant subspace \( M \)
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 self-adjoint operator | restricted to M closed invariant subspace ) * = equals A self-adjoint operator | restricted to M closed invariant subspace \(
\mathopen{}\left(A|M\right)\mathclose{}^{*}= A|M
\)
A self-adjoint operator ( M closed invariant subspace ⊥ ) ⊆ subset M closed invariant subspace ⊥ \(
A\mathopen{}\left( {M}^{\perp}\right)\mathclose{}\subseteq {M}^{\perp}
\)
( A self-adjoint operator | restricted to M closed invariant subspace ⊥ ) * = equals A self-adjoint operator | restricted to M closed invariant subspace ⊥ \(
\mathopen{}\left(A|{M}^{\perp}\right)\mathclose{}
^{*}= A|{M}^{\perp}
\)
〈 A self-adjoint operator ( x vector ) , y vector 〉 = equals 〈 x vector , A self-adjoint operator ( y vector ) 〉 \(
\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, A\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}
\) x vector \( x \) y vector \( y \) M closed invariant subspace \( M \)
x vector ∈ element of M closed invariant subspace ⊥ \(
x\in {M}^{\perp}
\)
y vector ∈ element of M closed invariant subspace \(
y\in M
\)
〈 A self-adjoint operator ( x vector ) , y vector 〉 = equals 〈 x vector , A self-adjoint operator ( y vector ) 〉 = equals 0 zero \(
\mathopen{}\left\langle{}A\mathopen{}\left( x\right)\mathclose{}, y\right\rangle\mathclose{}= \mathopen{}\left\langle{}x, A\mathopen{}\left( y\right)\mathclose{}\right\rangle\mathclose{}= 0
\) M closed invariant subspace \( M \)
y vector ∈ element of M closed invariant subspace \(
y\in M
\)
A self-adjoint operator ( y vector ) ∈ element of M closed invariant subspace \(
A\mathopen{}\left( y\right)\mathclose{}\in M
\)
Follows from the two items above.
Theorem II.60
Let
A self-adjoint operator = equals A self-adjoint operator * ∈ element of 𝒦 compact linear operators ( H Hilbert space ) \(
A= A^{*}\in \mathcal{K}\mathopen{}\left( H\right)\mathclose{}
\) H Hilbert space \( H \) A self-adjoint operator \( A \)
Let M closed subspace \( M \)
Ker kernel ( A self-adjoint operator ) \(
\operatorname{Ker}\mathopen{}\left( A\right)\mathclose{}
\)
⋃ union
λ complex number ∈ element of σ ( A self-adjoint operator ) ∖ set difference { set 0 zero } set
Ker kernel ( λ complex number - minus A self-adjoint operator ) \(
\bigcup_{
λ\in \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\setminus \mathopen{}\left\{\, 0\,\right\}\mathclose{}
}{}
\operatorname{Ker}\mathopen{}\left( λ-A\right)\mathclose{}
\) Lemma II.58 plainly does have an
orthonormal basis of A self-adjoint operator \( A \)
A self-adjoint operator ( M closed subspace ) ⊆ subset M closed subspace \(
A\mathopen{}\left( M\right)\mathclose{}\subseteq M
\) Lemma II.59 that
A self-adjoint operator | restricted to M closed subspace ⊥ \(
A|{M}^{\perp}
\)
M closed subspace ⊥ \(
{M}^{\perp}
\)
M closed subspace ⊥ ≠ not equal to 0 zero \(
{M}^{\perp}\neq 0
\)
A self-adjoint operator | restricted to M closed subspace ⊥ ≠ not equal to 0 zero \(
A|{M}^{\perp}\neq 0
\)
M closed subspace ⊥ ⊆ subset Ker kernel ( A self-adjoint operator ) ⊆ subset M closed subspace \(
{M}^{\perp}\subseteq \operatorname{Ker}\mathopen{}\left( A\right)\mathclose{}\subseteq M
\)
‖ A self-adjoint operator | restricted to M closed subspace ⊥ ‖ = equals ρ Euclidean norm ( A self-adjoint operator | restricted to M closed subspace ⊥ ) \(
\mathopen{}\left\lVert{}A|{M}^{\perp}\right\rVert\mathclose{}= ρ\mathopen{}\left( A|{M}^{\perp}\right)\mathclose{}
\)
A self-adjoint operator | restricted to M closed subspace ⊥ ≠ not equal to 0 zero \(
A|{M}^{\perp}\neq 0
\)
A self-adjoint operator | restricted to M closed subspace ⊥ \(
A|{M}^{\perp}
\) μ measure \( μ \)
Ker kernel ( μ measure - minus A self-adjoint operator ) ⊆ subset M closed subspace \(
\operatorname{Ker}\mathopen{}\left( μ-A\right)\mathclose{}\subseteq M
\)
Ker kernel ( μ measure - minus A self-adjoint operator ) ∩ intersection M closed subspace ⊥ = equals { set 0 zero } set \(
\operatorname{Ker}\mathopen{}\left( μ-A\right)\mathclose{}\cap {M}^{\perp}= \mathopen{}\left\{\, 0\,\right\}\mathclose{}
\)
M closed subspace = equals H Hilbert space \(
M= H
\)