Spectral Theorem for compact self-adjoint operators

Spectral Theorem for compact self-adjoint operators

Philippe Jaming

Universit ´e de Bordeaux

http://www.u-bordeaux.fr/˜ pjaming/enseignement/M1.html

Master Math ´ematiques et Applications

M1 Math ´ematiques fondamentales & M1 Analyse, EDP, probabilit ´es Lecture : Introduction to spectral analysis

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 1 / 14

Preliminary warning

This video is a complement to the lecture notes available at

http://www.u-bordeaux.fr/˜ pjaming/enseignement/M1.html

1. Statement of the Spectral Theorem 2. An example

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 2 / 14

Preliminary warning

This video is a complement to the lecture notes available at

http://www.u-bordeaux.fr/˜ pjaming/enseignement/M1.html

1. Statement of the Spectral Theorem 2. An example

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 2 / 14

Preliminary warning

This video is a complement to the lecture notes available at

http://www.u-bordeaux.fr/˜ pjaming/enseignement/M1.html

1. Statement of the Spectral Theorem 2. An example

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 2 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space.T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

Spectral Theorem for compact self-adjoint operators

Aim : give an example which illustrates the spectral theorem Theorem (Spectral Theorem)

H separable, infinite dimnesional, Hilbert space. T :H →Hcompact, self-adjointoperator.

∃(e_k)k∈Northonormal basis of H ;

∃(λ_k)k∈Nreal,λ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Difficulty : no characteristic polynomial to find eigenvectors

Necessary to develop strategies which will depend on the (family of) operators considered.

Here : convolutions onL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 14

The operator

Hilbert space :H =L²(T)

={f :R→C, f 1-periodic,kfk²₂=R1

0 |f(t)|²dt <+∞}.

Operator :T :L²→L²,Tf(x) =H∗f(x) = Z 1

0

f(t)H(x−t)dtwith

H(t) =







0 sit∈(−1/2,0) 1 sit∈(0,1/2) 1/2 ift =0 ort=1/2

.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 14

The operator

Hilbert space :H =L²(T)

={f :R→C, f 1-periodic,kfk²₂=R1

0 |f(t)|²dt <+∞}.

Operator :T :L²→L²,Tf(x) =H∗f(x) = Z 1

0

f(t)H(x−t)dtwith

H(t) =







0 sit∈(−1/2,0) 1 sit∈(0,1/2) 1/2 ift =0 ort=1/2

.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 14

The operator

Hilbert space :H =L²(T)

={f :R→C, f 1-periodic,kfk²₂=R1

0 |f(t)|²dt <+∞}.

Operator :T :L²→L²,Tf(x) =H∗f(x) = Z 1

0

f(t)H(x−t)dtwith

H(t) =







0 sit∈(−1/2,0) 1 sit∈(0,1/2) 1/2 ift =0 ort=1/2

.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 14

The operator

Hilbert space :H =L²(T)

={f :R→C, f 1-periodic,kfk²₂=R1

0 |f(t)|²dt <+∞}.

Operator :T :L²→L²,Tf(x) =H∗f(x)= Z 1

0

f(t)H(x−t)dtwith

H(t) =







0 sit∈(−1/2,0) 1 sit∈(0,1/2) 1/2 ift =0 ort=1/2

.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 14

The operator

Hilbert space :H =L²(T)

={f :R→C, f 1-periodic,kfk²₂=R1

0 |f(t)|²dt <+∞}.

Operator :T :L²→L²,Tf(x) =H∗f(x) = Z 1

0

f(t)H(x−t)dtwith

H(t) =







0 sit∈(−1/2,0) 1 sit∈(0,1/2) 1/2 ift =0 ort=1/2

.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 14

The operator

Hilbert space :H =L²(T)

={f :R→C, f 1-periodic,kfk²₂=R1

0 |f(t)|²dt <+∞}.

Operator :T :L²→L²,Tf(x) =H∗f(x) = Z 1

0

f(t)H(x−t)dtwith

H(t) =







0 sit∈(−1/2,0) 1 sit∈(0,1/2) 1/2 ift =0 ort=1/2

.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is self-adjoint

hTf,gi= Z 1

0

Tf(x)g(x)dx = Z 1

0

Z 1 0

f(t)H(x−t)dtg(x)dx Fubini

= Z 1

0

Z 1 0

f(t)H(x−t)g(x)dxdt= Z 1

0

f(t) Z 1

0

H(x−t)g(x)dx

! dt

= Z 1

0

f(t) Z 1

0

H(x −t)g(x)dx

! dt=

Z 1 0

f(t)H∗g(t)dt=hf,Tgi.

We are allowed to use Fubini :Cauchy-Schwarz Z 1

0

Z 1 0

|f(t)H(x−t)g(x)|dxdt≤

Z 1 0

|f(t)|

Z 1 0

|H(x−t)|²dt

!1/2

kgk₂dt≤R1

0 |f(t)|kHk₂kgk₂dt<+∞.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 14

T is compact - Method 1

Tf(x) = Z 1

0

K(x,t)f(t)dt withK(x,t) =H(x −t).

Z 1 0

Z 1 0

|K(x,t)|²dxdt = Z 1

0

Z 1 0

|H(x −t)|²dxdt= Z 1

0

kHk²₂dt

=kHk²₂<+∞.

T is Hilbert-Schmidt thuscompact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 14

T is compact - Method 1

Tf(x) = Z 1

0

K(x,t)f(t)dt withK(x,t) =H(x −t).

Z 1 0

Z 1 0

|K(x,t)|²dxdt = Z 1

0

Z 1 0

|H(x −t)|²dxdt= Z 1

0

kHk²₂dt

=kHk²₂<+∞.

T is Hilbert-Schmidt thuscompact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 14

T is compact - Method 1

Tf(x) = Z 1

0

K(x,t)f(t)dt withK(x,t) =H(x −t).

Z 1 0

Z 1 0

|K(x,t)|²dxdt = Z 1

0

Z 1 0

|H(x −t)|²dxdt= Z 1

0

kHk²₂dt

=kHk²₂<+∞.

T is Hilbert-Schmidt thuscompact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 14

T is compact - Method 1

Tf(x) = Z 1

0

K(x,t)f(t)dt withK(x,t) =H(x −t).

Z 1 0

Z 1 0

|K(x,t)|²dxdt = Z 1

0

Z 1 0

|H(x −t)|²dxdt= Z 1

0

kHk²₂dt

=kHk²₂<+∞.

T is Hilbert-Schmidt thuscompact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 14

T is compact - Method 1

Tf(x) = Z 1

0

K(x,t)f(t)dt withK(x,t) =H(x −t).

Z 1 0

Z 1 0

|K(x,t)|²dxdt = Z 1

0

Z 1 0

|H(x −t)|²dxdt= Z 1

0

kHk²₂dt

=kHk²₂<+∞.

T is Hilbert-Schmidt thuscompact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 14

T is compact - Method 1

Tf(x) = Z 1

0

K(x,t)f(t)dt withK(x,t) =H(x −t).

Z 1 0

Z 1 0

|K(x,t)|²dxdt = Z 1

0

Z 1 0

|H(x −t)|²dxdt= Z 1

0

kHk²₂dt

=kHk²₂<+∞.

T is Hilbert-Schmidtthuscompact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 14

T is compact - Method 1

Tf(x) = Z 1

0

K(x,t)f(t)dt withK(x,t) =H(x −t).

Z 1 0

Z 1 0

|K(x,t)|²dxdt = Z 1

0

Z 1 0

|H(x −t)|²dxdt= Z 1

0

kHk²₂dt

=kHk²₂<+∞.

T is Hilbert-Schmidt thuscompact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

Theorem (Arz ´ela-Ascoli)

K compact metric and X ⊂ C(K)is compact⇔

1 X is pointwise bounded :∀x ∈K ,∃M_x >0s.t.∀f ∈X ,|f(x)| ≤Mx.

2 X is equi-continuous :∀ε >0, x ∈K ,∃η_x s.t. d(x,y)< ηx ⇒

∀f ∈X ,|f(x)−f(y)|< ε.

3 X is closed.

K ⊂ C(T)(continuous 1-periodic) compact⇒K ⊂L²(T)andK compact inL²(T):

(f_n)_n⊂K extract(f_nk)_k uniformly convergent sequence⇒(f_nk)_k convergent inL²(T).

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 14

T is compact - Method 2 : Ascoli-Arzela

f,H ∈L²(T)thusf∗H ∈ C([0,1]).

–[0,1]is compact ;

– ifkfk₂≤1 then (Cauchy-Schwarz)|H∗f(x)|=

R1

0 H(x −t)f(t)dt ≤ R1

0 |H(x −t)|²dt 1/2

kfk₂≤ kHk₂(Cauchy-Schwarz) ; T(B)ispointwise bounded;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 14

T is compact - Method 2 : Ascoli-Arzela

f,H ∈L²(T)thusf∗H ∈ C([0,1]).

–[0,1]is compact ;

– ifkfk₂≤1 then (Cauchy-Schwarz)|H∗f(x)|=

R1

0 H(x −t)f(t)dt ≤ R1

0 |H(x −t)|²dt 1/2

kfk₂≤ kHk₂(Cauchy-Schwarz) ; T(B)ispointwise bounded;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 14

T is compact - Method 2 : Ascoli-Arzela

f,H ∈L²(T)thusf∗H ∈ C([0,1]).

–[0,1]is compact ;

– ifkfk₂≤1 then (Cauchy-Schwarz)|H∗f(x)|=

R1

0 H(x −t)f(t)dt ≤ R1

0 |H(x −t)|²dt 1/2

kfk₂≤ kHk₂(Cauchy-Schwarz) ; T(B)ispointwise bounded;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 14

T is compact - Method 2 : Ascoli-Arzela

f,H ∈L²(T)thusf∗H ∈ C([0,1]).

–[0,1]is compact ;

– ifkfk₂≤1 then (Cauchy-Schwarz)|H∗f(x)|=

R1

0 H(x −t)f(t)dt ≤ R1

0 |H(x −t)|²dt 1/2

kfk₂≤ kHk₂(Cauchy-Schwarz) ; T(B)ispointwise bounded;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 14

T is compact - Method 2 : Ascoli-Arzela

f,H ∈L²(T)thusf∗H ∈ C([0,1]).

–[0,1]is compact ;

– ifkfk₂≤1 then (Cauchy-Schwarz)|H∗f(x)|=

R1

0 H(x −t)f(t)dt ≤ R1

0 |H(x −t)|²dt 1/2

kfk₂≤ kHk₂(Cauchy-Schwarz) ; T(B)ispointwise bounded;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 2 : Ascoli-Arzela

– ifkfk₂≤1 thenH∗f(x) =R1

0 τxH(t)f(t)ˇ dt whereHˇ(s) =H(−s)τxϕ(s) =ϕ(s−x).

|H∗f(x)−H∗f(y)|=

Z 1 0

τxH(t)ˇ −τyH(t)ˇ f(t)dt

≤

τxHˇ −τyHˇ 2kfk₂

with Cauchy-Schwarz.|Tf(x)−Tf(y)| ≤

τxHˇ −τyHˇ

₂→0 when x →y,

T(B)isequicontinuous.

Arzela-Ascoli⇒T(B)is compact inC([0,1])

⇒T(B)is compact inL²(T)

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 14

T is compact - Method 3 : Kolmogorov-Riesz

Theorem (Kolmogorov-Riesz) X ⊂L²(R)is compact⇔

i X is closed ;

ii X is bounded :∃M>0∀f ∈X ,kfk₂≤M ;

iii X is equi-integrable :∀ε >0∃R >0,∀f ∈X ,R

|x|≥R|f(x)|²dx ≤ε.

iv Translations are equi-continuous on X :∀ε >0,∃ρ >0s.t.|y|< ρ

⇒ ∀f ∈X ,

f−τyf ₂≤ε.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 14

T is compact - Method 3 : Kolmogorov-Riesz

Theorem (Kolmogorov-Riesz) X ⊂L²(R)is compact⇔

i X is closed ;

ii X is bounded :∃M>0∀f ∈X ,kfk₂≤M ;

iii X is equi-integrable :∀ε >0∃R >0,∀f ∈X ,R

|x|≥R|f(x)|²dx ≤ε.

iv Translations are equi-continuous on X :∀ε >0,∃ρ >0s.t.|y|< ρ

⇒ ∀f ∈X ,

f−τyf ₂≤ε.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 14

T is compact - Method 3 : Kolmogorov-Riesz

Theorem (Kolmogorov-Riesz) X ⊂L²(R)is compact⇔

i X is closed ;

ii X is bounded :∃M>0∀f ∈X ,kfk₂≤M ;

iii X is equi-integrable :∀ε >0∃R >0,∀f ∈X ,R

|x|≥R|f(x)|²dx ≤ε.

iv Translations are equi-continuous on X :∀ε >0,∃ρ >0s.t.|y|< ρ

⇒ ∀f ∈X ,

f−τyf ₂≤ε.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 14

T is compact - Method 3 : Kolmogorov-Riesz

Theorem (Kolmogorov-Riesz) X ⊂L²(R)is compact⇔

i X is closed ;

ii X is bounded :∃M>0∀f ∈X ,kfk₂≤M ;

iii X is equi-integrable :∀ε >0∃R >0,∀f ∈X ,R

|x|≥R|f(x)|²dx ≤ε.

iv Translations are equi-continuous on X :∀ε >0,∃ρ >0s.t.|y|< ρ

⇒ ∀f ∈X ,

f−τyf ₂≤ε.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 14

T is compact - Method 3 : Kolmogorov-Riesz

Theorem (Kolmogorov-Riesz) X ⊂L²(R)is compact⇔

i X is closed ;

ii X is bounded :∃M>0∀f ∈X ,kfk₂≤M ;

iii X is equi-integrable :∀ε >0∃R >0,∀f ∈X ,R

|x|≥R|f(x)|²dx ≤ε.

iv Translations are equi-continuous on X :∀ε >0,∃ρ >0s.t.|y|< ρ

⇒ ∀f ∈X ,

f−τyf ₂≤ε.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 14

T is compact - Method 3 : Kolmogorov-Riesz

Theorem (Kolmogorov-Riesz) X ⊂L²(R)is compact⇔

i X is closed ;

ii X is bounded :∃M>0∀f ∈X ,kfk₂≤M ;

iii X is equi-integrable :∀ε >0∃R >0,∀f ∈X ,R

|x|≥R|f(x)|²dx ≤ε.

iv Translations are equi-continuous on X :∀ε >0,∃ρ >0s.t.|y|< ρ

⇒ ∀f ∈X ,

f−τyf ₂≤ε.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 14