Lecture Notes in Functional Analysis
by William L. Paschke
edition 0.9
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E. Integral Operators
Let S Volterra operator \( S \)
L 2 Lebesgue space ( [ interval 0 zero , 1 one ] interval ) = equals H Hilbert space \(
\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}= H
\)
( S Volterra operator ( f function ) ) ( x real number ) = equals ∫ integral 0 zero x real number f function \(
\mathopen{}\left(S\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= \int _{0}^{x}{}f
\)
S Volterra operator 2 two \(
{S}^{2}
\)
( S Volterra operator 2 two ( f function ) ) ( x real number ) = equals ∫ integral 0 zero x real number
( S Volterra operator ( f function ) ) ( s real number ) d s real number = equals ∫ integral 0 zero x real number
∫ integral 0 zero s real number
f function ( t real number ) d t real number d s real number . \[
\mathopen{}\left({S}^{2}\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= \int _{0}^{x}{}
\mathopen{}\left(S\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( s\right)\mathclose{}
\,\mathrm{d}s= \int _{0}^{x}{}
\int _{0}^{s}{}
f\mathopen{}\left( t\right)\mathclose{}
\,\mathrm{d}t
\,\mathrm{d}s
\text{.}
\]
( S Volterra operator 2 two ( f function ) ) ( x real number ) = equals ∫ integral 0 zero x real number
∫ integral t real number x real number f function ( t real number ) d s real number d t real number = equals ∫ integral 0 zero x real number
( x real number - minus t real number ) times f function ( t real number ) d t real number = equals ∫ integral 0 zero 1 one
τ function ( x real number t real number ) times f function ( t real number ) d t real number \[
\mathopen{}\left({S}^{2}\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= \int _{0}^{x}{}
\int _{t}^{x}{}f\mathopen{}\left( t\right)\mathclose{}\,\mathrm{d}s
\,\mathrm{d}t= \int _{0}^{x}{}
\mathopen{}\left(x-t\right)\mathclose{}f\mathopen{}\left( t\right)\mathclose{}
\,\mathrm{d}t= \int _{0}^{1}{}
τ\mathopen{}\left( x, t\right)\mathclose{}f\mathopen{}\left( t\right)\mathclose{}
\,\mathrm{d}t
\]
τ function ( x real number t real number ) = equals { cases x real number - minus t real number , 0 zero ≤ less than or equal to t real number ≤ less than or equal to x real number ; 0 zero , else, } \[
τ\mathopen{}\left( x, t\right)\mathclose{}= \begin{cases}x-t, & 0\leq t\leq x; \\ 0, & \text{else,}\end{cases}
\]
[ interval 0 zero , 1 one ] interval 2 two \(
{\mathopen{}\left[0, 1\right]\mathclose{}}^{2}
\)
We say that
S Volterra operator 2 two \(
{S}^{2}
\) integral operator with kernel τ function \( τ \)
H Hilbert space = equals L 2 Lebesgue space ( [ interval 0 zero , 1 one ] interval ) \(
H= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}
\)
[ interval 0 zero , 1 one ] interval \(
\mathopen{}\left[0, 1\right]\mathclose{}
\)
e unit vector 0 zero = equals 1 one \(
{e}_{0}= 1
\)
e unit vector n integer ( x real number ) = equals 2 two times cos cosine ( n integer times π times x real number ) \(
{e}_{n}\mathopen{}\left( x\right)\mathclose{}= \sqrt{2}\cos\mathopen{}\left( nπx\right)\mathclose{}
\)
n integer ∈ element of { set 1 one 2 two … } set \(
n\in \mathopen{}\left\{\, 1, 2, \dotsc\,\right\}\mathclose{}
\)
[ interval 0 zero , 1 one ] interval \(
\mathopen{}\left[0, 1\right]\mathclose{}
\)
[ interval − 1 one , 1 one ] interval \(
\mathopen{}\left[{-}1, 1\right]\mathclose{}
\)
{ set e unit vector 0 zero e unit vector 1 one e unit vector 2 two … } set \(
\mathopen{}\left\{\, {e}_{0}, {e}_{1}, {e}_{2}, \dotsc\,\right\}\mathclose{}
\) H Hilbert space \( H \)
[ interval 0 zero , 1 one ] interval \(
\mathopen{}\left[0, 1\right]\mathclose{}
\)
[ interval − 1 one , 1 one ] interval \(
\mathopen{}\left[{-}1, 1\right]\mathclose{}
\) 1 one \( 1 \)
− 1 one \(
{-}1
\)
Let
P projection n integer \(
{P}_{n}
\)
span span ( e unit vector 0 zero e unit vector 1 one … e unit vector n integer ) \(
\operatorname{span}\mathopen{}\left( {e}_{0}, {e}_{1}, \dotsc, {e}_{n}\right)\mathclose{}
\)
P projection n integer ( f function ) \(
{P}_{n}\mathopen{}\left( f\right)\mathclose{}
\) n integer \( n \) f function \( f \)
∑ summation n integer = 0 zero ∞ infinity
〈 f function , e unit vector n integer 〉 times e unit vector n integer . \[
\sum_{n=0}^{\infty}{}
\mathopen{}\left\langle{}f, {e}_{n}\right\rangle\mathclose{}{e}_{n}
\text{.}
\]
‖ P projection n integer ( f function ) - minus f function ‖ ∞ infinity \(
\mathopen{}\left\lVert{}{P}_{n}\mathopen{}\left( f\right)\mathclose{}-f\right\rVert\mathclose{}_{\infty}
\) f function \( f \)
C Cesàro mean n integer = equals 1 one n integer + plus 1 one times ( P projection 0 zero + plus P projection 1 one + plus ⋯ + plus P projection n integer ) \(
{C}_{n}= \frac{1}{n+1}\mathopen{}\left({P}_{0}+{P}_{1}+\dotsb+{P}_{n}\right)\mathclose{}
\)
Theorem III.41 (Fejer's Theorem)
Let F family of functions \( F \)
[ interval 0 zero , 1 one ] interval \(
\mathopen{}\left[0, 1\right]\mathclose{}
\)
ε positive real number > greater than 0 zero \(
ε\gt 0
\) N integer \( N \)
‖ C Cesàro mean n integer ( f function ) - minus f function ‖ ∞ infinity < less than ε positive real number \(
\mathopen{}\left\lVert{}{C}_{n}\mathopen{}\left( f\right)\mathclose{}-f\right\rVert\mathclose{}_{\infty}\lt ε
\)
n integer > greater than N integer \(
n\gt N
\)
f function ∈ element of F family of functions \(
f\in F
\)
C Cesàro mean n integer ( f function ) → converges to f function \(
{C}_{n}\mathopen{}\left( f\right)\mathclose{} \to f
\)
f function ∈ element of C space of continuous functions ( [ interval 0 zero , 1 one ] interval ) \(
f\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}
\)
See for instance Chapter 5 of Young [12 ] for the proof in the
standard situation of a single
2 two times π \( 2π \) g function \( g \) R real numbers \( \mathbb{R} \) g function \( g \)
‖ g function ‖ ∞ infinity \(
\mathopen{}\left\lVert{}g\right\rVert\mathclose{}_{\infty}
\)
| modulus g function ( x real number ) - minus g function ( y real number ) | modulus \(
\mathopen{}\left\lvert{}g\mathopen{}\left( x\right)\mathclose{}-g\mathopen{}\left( y\right)\mathclose{}\right\rvert\mathclose{}
\)
| modulus x real number - minus y real number | modulus \(
\mathopen{}\left\lvert{}x-y\right\rvert\mathclose{}
\)
Theorem III.42
Suppose
T trace-class operator ∈ element of 𝒯 trace-class operators ( H Hilbert space ) \(
T\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}
\)
T trace-class operator ( f function ( x real number ) ) = equals ∫ integral 0 zero 1 one
τ continuous function ( x real number t real number ) times f function ( t real number ) d t real number \[
T\mathopen{}\left( f\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}= \int _{0}^{1}{}
τ\mathopen{}\left( x, t\right)\mathclose{}f\mathopen{}\left( t\right)\mathclose{}
\,\mathrm{d}t
\] τ continuous function \( τ \)
[ interval 0 zero , 1 one ] interval 2 two \(
{\mathopen{}\left[0, 1\right]\mathclose{}}^{2}
\) T trace-class operator \( T \)
Tr trace ( T trace-class operator ) = equals ∫ integral 0 zero 1 one τ continuous function ( x real number x real number ) d x real number \(
\operatorname{Tr}\mathopen{}\left( T\right)\mathclose{}= \int _{0}^{1}{}τ\mathopen{}\left( x, x\right)\mathclose{}\,\mathrm{d}x
\)
This proof is taken from Gohberg [3 ] . With
( sequence e unit vector n integer ) sequence n integer = 0 zero ∞ infinity \(
\mathopen{}\left({e}_{n}\right)\mathclose{}_{n=0}^{\infty}
\)
Tr trace ( T trace-class operator ) = equals ∑ summation n integer
〈 T trace-class operator ( e unit vector n integer ) , e unit vector n integer 〉 \(
\operatorname{Tr}\mathopen{}\left( T\right)\mathclose{}= \sum_{n}{}
\mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{}
\)
τ continuous function x real number ∈ element of C space of continuous functions ( [ interval 0 zero , 1 one ] interval ) \(
{τ}_{x}\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}
\)
τ continuous function x real number ( t real number ) = equals τ continuous function ( x real number t real number ) \(
{τ}_{x}\mathopen{}\left( t\right)\mathclose{}= τ\mathopen{}\left( x, t\right)\mathclose{}
\)
〈 T trace-class operator ( e unit vector n integer ) , e unit vector n integer 〉 = equals ∫ integral 0 zero 1 one
( T trace-class operator ( e unit vector n integer ) ) ( x real number ) times e unit vector n integer ( x real number ) d x real number = equals ∫ integral 0 zero 1 one
∫ integral 0 zero 1 one
τ continuous function ( x real number t real number ) times e unit vector n integer ( t real number ) times e unit vector n integer ( x real number ) d t real number d x real number = equals ∫ integral 0 zero 1 one
( 〈 τ continuous function x real number , e unit vector n integer 〉 times e unit vector n integer ) ( x real number ) d x real number . \[
\mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{}= \int _{0}^{1}{}
\mathopen{}\left(T\mathopen{}\left( {e}_{n}\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}{e}_{n}\mathopen{}\left( x\right)\mathclose{}
\,\mathrm{d}x= \int _{0}^{1}{}
\int _{0}^{1}{}
τ\mathopen{}\left( x, t\right)\mathclose{}{e}_{n}\mathopen{}\left( t\right)\mathclose{}{e}_{n}\mathopen{}\left( x\right)\mathclose{}
\,\mathrm{d}t
\,\mathrm{d}x= \int _{0}^{1}{}
\mathopen{}\left(\mathopen{}\left\langle{}{τ}_{x}, {e}_{n}\right\rangle\mathclose{}{e}_{n}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}
\,\mathrm{d}x
\text{.}
\]
A Cesàro mean n integer \(
{A}_{n}
\) n integer \( n \)
∑ summation
〈 T trace-class operator ( e unit vector n integer ) , e unit vector n integer 〉 \(
\sum{}
\mathopen{}\left\langle{}T\mathopen{}\left( {e}_{n}\right)\mathclose{}, {e}_{n}\right\rangle\mathclose{}
\)
A Cesàro mean n integer → converges to Tr trace ( T trace-class operator ) \(
{A}_{n} \to \operatorname{Tr}\mathopen{}\left( T\right)\mathclose{}
\)
A Cesàro mean n integer = equals ∫ integral 0 zero 1 one
( C Cesàro mean n integer ( τ continuous function x real number ) ) ( x real number ) d x real number \(
{A}_{n}= \int _{0}^{1}{}
\mathopen{}\left({C}_{n}\mathopen{}\left( {τ}_{x}\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}
\,\mathrm{d}x
\)
| modulus A Cesàro mean n integer - minus ∫ integral 0 zero 1 one τ continuous function ( x real number x real number ) d x real number | modulus = equals | modulus ∫ integral 0 zero 1 one
( ( C Cesàro mean n integer ( τ continuous function x real number ) ) ( x real number ) - minus τ continuous function x real number ( x real number ) ) d x real number | modulus ≤ less than or equal to ∫ integral 0 zero 1 one
‖ C Cesàro mean n integer ( τ continuous function x real number ) - minus τ continuous function x real number ‖ ∞ infinity d x real number ≤ less than or equal to sup supremum x real number
‖ C Cesàro mean n integer ( τ continuous function x real number ) - minus τ continuous function x real number ‖ → converges to 0 zero \[
\mathopen{}\left\lvert{}{A}_{n}-\int _{0}^{1}{}τ\mathopen{}\left( x, x\right)\mathclose{}\,\mathrm{d}x\right\rvert\mathclose{}= \mathopen{}\left\lvert{}\int _{0}^{1}{}
\mathopen{}\left(\mathopen{}\left({C}_{n}\mathopen{}\left( {τ}_{x}\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}-{τ}_{x}\mathopen{}\left( x\right)\mathclose{}\right)\mathclose{}
\,\mathrm{d}x\right\rvert\mathclose{}\leq \int _{0}^{1}{}
\mathopen{}\left\lVert{}{C}_{n}\mathopen{}\left( {τ}_{x}\right)\mathclose{}-{τ}_{x}\right\rVert\mathclose{}_{\infty}
\,\mathrm{d}x\leq \sup_{x}{}
\mathopen{}\left\lVert{}{C}_{n}\mathopen{}\left( {τ}_{x}\right)\mathclose{}-{τ}_{x}\right\rVert\mathclose{}
\to 0
\] Fejer for Families because
F = equals { set τ continuous function x real number | such that
x real number ∈ element of [ interval 0 zero , 1 one ] interval } set \(
F= \mathopen{}\left\{\, {τ}_{x}\,\middle\vert\,
, x\in \mathopen{}\left[0, 1\right]\mathclose{},
\,\right\}\mathclose{}
\) τ continuous function \( τ \)
In particular, if S Volterra operator \( S \)
Tr trace ( S Volterra operator 2 two ) = equals 0 zero \(
\operatorname{Tr}\mathopen{}\left( {S}^{2}\right)\mathclose{}= 0
\)
Corollary III.43
For any
r function ∈ element of C space of continuous functions ( [ interval 0 zero , 1 one ] interval ) \(
r\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}
\) T trace-class operator \( T \)
( T trace-class operator ( f function ) ) ( x real number ) = equals ∫ integral 0 zero x real number
( x real number - minus t real number ) times f function ( t real number ) times r function ( t real number ) d t real number \[
\mathopen{}\left(T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= \int _{0}^{x}{}
\mathopen{}\left(x-t\right)\mathclose{}f\mathopen{}\left( t\right)\mathclose{}r\mathopen{}\left( t\right)\mathclose{}
\,\mathrm{d}t
\]
Tr trace ( T trace-class operator ) = equals 0 zero \(
\operatorname{Tr}\mathopen{}\left( T\right)\mathclose{}= 0
\)
If S Volterra operator \( S \)
M r function multiplication operator r function \(
\mathrm{M}_{r}
\) r function \( r \)
T trace-class operator = equals S Volterra operator 2 two ∘ composition M r function multiplication operator r function ∈ element of 𝒯 trace-class operators ( H Hilbert space ) \(
T= {S}^{2}\circ \mathrm{M}_{r}\in \mathcal{T}\mathopen{}\left( H\right)\mathclose{}
\) T trace-class operator \( T \)
τ function ( x real number t real number ) times r function ( t real number ) \(
τ\mathopen{}\left( x, t\right)\mathclose{}r\mathopen{}\left( t\right)\mathclose{}
\) 0 zero \( 0 \)