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

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

3. Toolbox 4. The proof

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 2 / 11

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

3. Toolbox 4. The proof

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

3. Toolbox 4. The proof

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 2 / 11

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

3. Toolbox 4. The proof

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

3. Toolbox 4. The proof

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 2 / 11

Spectral Theorem for compact self-adjoint operators

Aim: give the main tools to prove the spectral theorem Theorem (Spectral Theorem)

H , infinite dimensional, Hilbert space. T :H →Hcompact, self-adjoint, one-to-oneoperator.

∃(e_k)k∈Northonormal basis of H;

∃(λ_k)_k∈Nreal, ,λ_k decreasingλ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Spectral Theorem for compact self-adjoint operators

Aim: give the main tools to prove the spectral theorem Theorem (Spectral Theorem)

H , infinite dimensional, Hilbert space. T :H →Hcompact, self-adjoint, one-to-oneoperator.

∃(e_k)k∈Northonormal basis of H;

∃(λ_k)_k∈Nreal, ,λ_k decreasingλ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 11

Spectral Theorem for compact self-adjoint operators

Aim: give the main tools to prove the spectral theorem Theorem (Spectral Theorem)

H , infinite dimensional, Hilbert space. T :H →Hcompact, self-adjoint, one-to-oneoperator.

∃(e_k)k∈Northonormal basis of H;

∃(λ_k)_k∈Nreal, ,λ_k decreasingλ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Spectral Theorem for compact self-adjoint operators

Aim: give the main tools to prove the spectral theorem Theorem (Spectral Theorem)

H , infinite dimensional, Hilbert space. T :H →Hcompact, self-adjoint, one-to-oneoperator.

∃(e_k)k∈Northonormal basis of H;

∃(λ_k)_k∈Nreal,,λ_k decreasingλ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 11

Spectral Theorem for compact self-adjoint operators

Aim: give the main tools to prove the spectral theorem Theorem (Spectral Theorem)

H , infinite dimensional, Hilbert space. T :H →Hcompact, self-adjoint, one-to-oneoperator.

∃(e_k)k∈Northonormal basis of H;

∃(λ_k)_k∈Nreal, ,λ_k decreasingλ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Spectral Theorem for compact self-adjoint operators

Aim: give the main tools to prove the spectral theorem Theorem (Spectral Theorem)

H , infinite dimensional, Hilbert space. T :H →Hcompact, self-adjoint, one-to-oneoperator.

∃(e_k)k∈Northonormal basis of H;

∃(λ_k)_k∈Nreal, ,λ_k decreasingλ_k →0;

Tx =X

k∈N

λ_khx,e_kie_k.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 11

Spectral theorem - comments

T has a matrix of the form

kerT E₁ E₂ · · ·

e_1,1, . . . ,e_n1,1 e_1,2, . . . ,e_n2,2

kerT 0 0 0 · · ·

e_1,1 ... en₁,1

0

λ1 0

. ..

0 λ1

0 e_1,2

... e_n2,2

0

λ₂ 0

. ..

0 λ₂

0 0

. .. 0

. ..

0 . ..

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 11

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 11

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 11

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 11

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

First step

The proof is an induction

i F₁=E_kTk⊕E_−kTk

ii π₁^±⊥-projection onE_±kTkΠ₁=π₁⁺+π₁⁻projection onF₁;

iii m₁= dimE_kTk,m₂= dimE_−kTk etd₁=m₁+m₂≥1;

iv e₁, . . . ,e_m1 ONB ofE_kTk &e_m1+1, . . . ,e_d1 ONB ofE_−kTk e₁, . . . ,e_d1 ONB ofF₁;

v λ1=· · ·=λm1 =kTk,λm₁+1=· · ·=λ_d1 =−kTk.

T = Π1TΠ1

| {z }

Pd1

k=1λkhx,e_kie_k

+ (I−Π1)T(I−Π1)

| {z }

=T1

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 11

Spectral theorem - comments

T has a matrix of the form

E₁ E₂ · · ·

e₁, . . . ,en₁ e_n1+1, . . . ,en₂

e₁ ... en₁

kTk 0

. ..

0 −kTk

0

e_n1+1 ... en₂

λ₂ 0

. ..

0 λ₂

0

. .. 0

. ..

0 . ..

Second step

T₁= (I−Π₁)T(I−Π₁)self-adjoint,compact,kT₁k ≤ kTk.

Observation kT₁k<kTk

Proof: If not,∃x,kxk=1,T₁x =kTkx (orT₁x =−kTkx)

—(I−Π₁)x =x otherwisek(I−π₁)xk<kxkand then

kTk=kT₁xk=k(I−Π1)T(I−Π1)xk ≤ kTkk(I−Π1)xk<kTk

— sokTkx =T₁x = (I−Π₁)T(I−Π₁)x = (I−Π₁)Tx =Tx−Π₁Tx Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π₁Tx

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 11

Second step

T₁= (I−Π₁)T(I−Π₁)self-adjoint,compact,kT₁k ≤ kTk.

Observation kT₁k<kTk

Proof: If not,∃x,kxk=1,T₁x =kTkx (orT₁x =−kTkx)

—(I−Π₁)x =x otherwisek(I−π₁)xk<kxkand then

kTk=kT₁xk=k(I−Π1)T(I−Π1)xk ≤ kTkk(I−Π1)xk<kTk

— sokTkx =T₁x = (I−Π₁)T(I−Π₁)x = (I−Π₁)Tx =Tx−Π₁Tx Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π₁Tx

Second step

T₁= (I−Π₁)T(I−Π₁)self-adjoint,compact,kT₁k ≤ kTk.

Observation kT₁k<kTk

Proof: If not,∃x,kxk=1,T₁x =kTkx (orT₁x =−kTkx)

—(I−Π₁)x =x otherwisek(I−π₁)xk<kxkand then

kTk=kT₁xk=k(I−Π1)T(I−Π1)xk ≤ kTkk(I−Π1)xk<kTk

— sokTkx =T₁x = (I−Π₁)T(I−Π₁)x = (I−Π₁)Tx =Tx−Π₁Tx Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π₁Tx

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 11

Second step

T₁= (I−Π₁)T(I−Π₁)self-adjoint,compact,kT₁k ≤ kTk.

Observation kT₁k<kTk

Proof: If not,∃x,kxk=1,T₁x =kTkx (orT₁x =−kTkx)

—(I−Π₁)x =x otherwisek(I−π₁)xk<kxkand then

kTk=kT₁xk=k(I−Π1)T(I−Π1)xk ≤ kTkk(I−Π1)xk<kTk

— sokTkx =T₁x = (I−Π₁)T(I−Π₁)x = (I−Π₁)Tx =Tx−Π₁Tx Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π₁Tx

Second step

T₁= (I−Π₁)T(I−Π₁)self-adjoint,compact,kT₁k ≤ kTk.

Observation kT₁k<kTk

Proof: If not,∃x,kxk=1,T₁x =kTkx (orT₁x =−kTkx)

—(I−Π₁)x =x otherwisek(I−π₁)xk<kxkand then

kTk=kT₁xk=k(I−Π1)T(I−Π1)xk≤ kTkk(I−Π1)xk<kTk

— sokTkx =T₁x = (I−Π₁)T(I−Π₁)x = (I−Π₁)Tx =Tx−Π₁Tx Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π₁Tx

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 11

Second step

T₁= (I−Π₁)T(I−Π₁)self-adjoint,compact,kT₁k ≤ kTk.

Observation kT₁k<kTk

Proof: If not,∃x,kxk=1,T₁x =kTkx (orT₁x =−kTkx)

—(I−Π₁)x =x otherwisek(I−π₁)xk<kxkand then

kTk=kT₁xk=k(I−Π1)T(I−Π1)xk ≤ kTkk(I−Π1)xk<kTk

— sokTkx =T₁x = (I−Π₁)T(I−Π₁)x = (I−Π₁)Tx =Tx−Π₁Tx Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π₁Tx

Second step

T₁= (I−Π₁)T(I−Π₁)self-adjoint,compact,kT₁k ≤ kTk.

Observation kT₁k<kTk

Proof: If not,∃x,kxk=1,T₁x =kTkx (orT₁x =−kTkx)

—(I−Π₁)x =x otherwisek(I−π₁)xk<kxkand then

kTk=kT₁xk=k(I−Π1)T(I−Π1)xk ≤ kTkk(I−Π1)xk<kTk

— sokTkx =T₁x = (I−Π₁)T(I−Π₁)x = (I−Π₁)Tx =Tx−Π₁Tx Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π₁Tx

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 7 / 11

Second step

T₁= (I−Π₁)T(I−Π₁)self-adjoint,compact,kT₁k ≤ kTk.

Observation kT₁k<kTk

Proof: If not,∃x,kxk=1,T₁x =kTkx (orT₁x =−kTkx)

—(I−Π₁)x =x otherwisek(I−π₁)xk<kxkand then

kTk=kT₁xk=k(I−Π1)T(I−Π1)xk ≤ kTkk(I−Π1)xk<kTk

— sokTkx =T₁x = (I−Π₁)T(I−Π₁)x = (I−Π₁)Tx =Tx−Π₁Tx Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π₁Tx

Second step

(I−Π1)x =x &Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π1Tx – thus

kTxk²=kTk²kxk²+kΠ₁Txk² Π₁Tx =0Tx =kTkx i.e. x∈E_kTk a contradiction

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 11

Second step

(I−Π1)x =x &Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π1Tx – thus

kTxk²=kTk²kxk²+kΠ₁Txk² Π₁Tx =0Tx =kTkx i.e. x∈E_kTk a contradiction

Second step

(I−Π1)x =x &Tx = kTkx

| {z }

∈Im(I−Π₁)

+Π1Tx – thus

kTxk²=kTk²kxk²+kΠ₁Txk² Π₁Tx =0Tx =kTkx i.e. x∈E_kTk a contradiction

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 11

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

| {z }

=Tn

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 11

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

| {z }

=Tn

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 11

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

| {z }

=Tn

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 11

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

| {z }

=Tn

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 11

Induction

T =

n−1

X

k=1

Π_kTΠ_k+ I−

n−1

X

k=1

Π_k

!

T I−

n−1

X

k=1

Π_k

!

| {z }

=Tn−1

i F_n=E_kTn−1k⊕E_−kTn−1k

ii π_n^±⊥-projection onE_±kTn−1kΠ_n=π⁺_n +π_n⁻projection onF_n;

iii m2n−1= dimE_kTnk,m_2n= dimE_−kTnk etdn=m2n−1+m_2n≥1;

iv e_dn−1+1, . . . ,e_{dn−1+m2n−1} ONB ofE_kTnk &e_{dn−1+m2n−1+1}, . . . ,e_dn ONB ofE_−kTnke_dn−1+1, . . . ,e_dn ONB ofF₁;

v λ_dn−1+1=· · ·=λ_{dn−1+m2n−1} =kT_n−1k, λ_{dn−1+m2n−1+1}=· · ·=λ_dn =−kT_n−1k.

T =Pn

k=1Π_kTΠ_k + (I−

n

X

k=1

Π_k)T(I−

n

X

k=1

Π_k)

Conclusion

We want to show thatT =

+∞

X

k=1

Π_kTΠ_k that isT_n→0.

We havekT_nk<kT_n−1k(λ_k decreases).

WriteDn=d1+· · ·+dnand note that|λ_Dn|=· · ·=|λ_Dn+1−1|=kT_nk If not,(e_Dn)infinite orthogonal system soe_Dn *0 (weakly),thus Te_Dn →0 strongly (T compact).

|λ_Dn|=|λ_Dn|ke_Dnk=kTe_Dnk →0.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 11

Conclusion

We want to show thatT =

+∞

X

k=1

Π_kTΠ_k that isT_n→0.

We havekT_nk<kT_n−1k(λ_k decreases).

WriteDn=d1+· · ·+dnand note that|λ_Dn|=· · ·=|λ_Dn+1−1|=kT_nk If not,(e_Dn)infinite orthogonal system soe_Dn *0 (weakly),thus Te_Dn →0 strongly (T compact).

|λ_Dn|=|λ_Dn|ke_Dnk=kTe_Dnk →0.

Conclusion

We want to show thatT =

+∞

X

k=1

Π_kTΠ_k that isT_n→0.

We havekT_nk<kT_n−1k(λ_k decreases).

WriteDn=d1+· · ·+dnand note that|λ_Dn|=· · ·=|λ_Dn+1−1|=kT_nk If not,(e_Dn)infinite orthogonal system soe_Dn *0 (weakly),thus Te_Dn →0 strongly (T compact).

|λ_Dn|=|λ_Dn|ke_Dnk=kTe_Dnk →0.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 11

Conclusion

We want to show thatT =

+∞

X

k=1

Π_kTΠ_k that isT_n→0.

We havekT_nk<kT_n−1k(λ_k decreases).

WriteDn=d1+· · ·+dnand note that|λ_Dn|=· · ·=|λ_Dn+1−1|=kT_nk If not,(e_Dn)infinite orthogonal system soe_Dn *0 (weakly),thus Te_Dn →0 strongly (T compact).

|λ_Dn|=|λ_Dn|ke_Dnk=kTe_Dnk →0.

Conclusion

We want to show thatT =

+∞

X

k=1

Π_kTΠ_k that isT_n→0.

We havekT_nk<kT_n−1k(λ_k decreases).

WriteDn=d1+· · ·+dnand note that|λ_Dn|=· · ·=|λ_Dn+1−1|=kT_nk If not,(e_Dn)infinite orthogonal system soe_Dn *0 (weakly),thus Te_Dn →0 strongly (T compact).

|λ_Dn|=|λ_Dn|ke_Dnk=kTe_Dnk →0.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 11