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 / 13

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

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

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 2 / 13

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

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

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 2 / 13

Spectral Theorem for compact self-adjoint operators

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

H separable, infinite dimensional, Hilbert space. T : H → H compact, self-adjoint operator.

∃ (e _k ) k∈ N orthonormal basis of H;

∃ (λ _k ) _{k∈ N} real, λ _k → 0;

Tx = X

k∈ N

λ _k hx , e _k ie _k .

Spectral Theorem for compact self-adjoint operators

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

H separable, infinite dimensional, Hilbert space. T : H → H compact, self-adjoint operator.

∃ (e _k ) k∈ N orthonormal basis of H;

∃ (λ _k ) _{k∈ N} real, λ _k → 0;

Tx = X

k∈ N

λ _k hx , e _k ie _k .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 13

Spectral Theorem for compact self-adjoint operators

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

H separable, infinite dimensional, Hilbert space. T : H → H compact, self-adjoint operator.

∃ (e _k ) k∈ N orthonormal basis of H;

∃ (λ _k ) _{k∈ N} real, λ _k → 0;

Tx = X

k∈ N

λ _k hx , e _k ie _k .

Spectral Theorem for compact self-adjoint operators

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

H separable, infinite dimensional, Hilbert space. T : H → H compact, self-adjoint operator.

∃ (e _k ) k∈ N orthonormal basis of H;

∃ (λ _k ) _{k∈ N} real, λ _k → 0;

Tx = X

k∈ N

λ _k hx , e _k ie _k .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 3 / 13

Spectral Theorem for compact self-adjoint operators

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

H separable, infinite dimensional, Hilbert space. T : H → H compact, self-adjoint operator.

∃ (e _k ) k∈ N orthonormal basis of H;

∃ (λ _k ) _{k∈ N} real, λ _k → 0;

Tx = X

k∈ N

λ _k hx , e _k ie _k .

Stability

Throughout H=Hilbert space, T : H → H bounded linear.

Observation

If T = T ^∗ and E ⊂ H closed subspace invariant through T : T (E ) ⊂ E then E ^⊥ is also stable through T

Proof.

x ∈ E , y ∈ E ^⊥ then Tx ∈ E and

hTy , x i =

* y

|{z}

∈E

, Tx

|{z}

∈E

+

= 0

that is Ty ∈ E ^⊥ .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 13

Stability

Throughout H=Hilbert space, T : H → H bounded linear.

Observation

If T = T ^∗ and E ⊂ H closed subspace invariant through T : T (E ) ⊂ E then E ^⊥ is also stable through T

Proof.

x ∈ E , y ∈ E ^⊥ then Tx ∈ E and

hTy , x i =

* y

|{z}

∈E

, Tx

|{z}

∈E

+

= 0

that is Ty ∈ E ^⊥ .

Stability

Throughout H=Hilbert space, T : H → H bounded linear.

Observation

If T = T ^∗ and E ⊂ H closed subspace invariant through T : T (E ) ⊂ E then E ^⊥ is also stable through T

Proof.

x ∈ E , y ∈ E ^⊥ then Tx ∈ E and

hTy , x i =

* y

|{z}

∈E

, Tx

|{z}

∈E

+

= 0

that is Ty ∈ E ^⊥ .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 13

Stability

Throughout H=Hilbert space, T : H → H bounded linear.

Observation

If T = T ^∗ and E ⊂ H closed subspace invariant through T : T (E ) ⊂ E then E ^⊥ is also stable through T

Proof.

x ∈ E , y ∈ E ^⊥ then Tx ∈ E and

hTy , x i =

* y

|{z}

∈E

, Tx

|{z}

∈E

+

= 0

that is Ty ∈ E ^⊥ .

Stability

Throughout H=Hilbert space, T : H → H bounded linear.

Observation

If T = T ^∗ and E ⊂ H closed subspace invariant through T : T (E ) ⊂ E then E ^⊥ is also stable through T

Proof.

x ∈ E , y ∈ E ^⊥ then Tx ∈ E and

hTy , x i =

* y

|{z}

∈E

, Tx

|{z}

∈E

+

= 0

that is Ty ∈ E ^⊥ .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 4 / 13

Stability

Throughout H=Hilbert space, T : H → H bounded linear.

Observation

If T = T ^∗ and E ⊂ H closed subspace invariant through T : T (E ) ⊂ E then E ^⊥ is also stable through T

Proof.

x ∈ E , y ∈ E ^⊥ then Tx ∈ E and

hTy , x i =

* y

|{z}

∈E

, Tx

|{z}

∈E

+

= 0

that is Ty ∈ E ^⊥ .

Kernel and image

ker T is stable through T . Observation

If T = T ^∗ , H = ker T ⊕ Im T , i.e. ker T = Im T ^⊥ i.e. (ker T ) ^⊥ = Im T . Proof.

x ∈ Im T ^⊥ means ∀y ∈ H

0 = hx , Ty i = hTx , y i

⇔ Tx = 0 i.e. x ∈ ker T .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 13

Kernel and image

ker T is stable through T . Observation

If T = T ^∗ , H = ker T ⊕ Im T , i.e. ker T = Im T ^⊥ i.e. (ker T ) ^⊥ = Im T . Proof.

x ∈ Im T ^⊥ means ∀y ∈ H

0 = hx , Ty i = hTx , y i

⇔ Tx = 0 i.e. x ∈ ker T .

Kernel and image

ker T is stable through T . Observation

If T = T ^∗ , H = ker T ⊕ Im T , i.e. ker T = Im T ^⊥ i.e. (ker T ) ^⊥ = Im T . Proof.

x ∈ Im T ^⊥ means ∀y ∈ H

0 = hx , Ty i = hTx , y i

⇔ Tx = 0 i.e. x ∈ ker T .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 13

Kernel and image

ker T is stable through T . Observation

If T = T ^∗ , H = ker T ⊕ Im T , i.e. ker T = Im T ^⊥ i.e. (ker T ) ^⊥ = Im T . Proof.

x ∈ Im T ^⊥ means ∀y ∈ H

0 = hx , Ty i = hTx , y i

⇔ Tx = 0 i.e. x ∈ ker T .

Kernel and image

ker T is stable through T . Observation

If T = T ^∗ , H = ker T ⊕ Im T , i.e. ker T = Im T ^⊥ i.e. (ker T ) ^⊥ = Im T . Proof.

x ∈ Im T ^⊥ means ∀y ∈ H

0 = hx , Ty i = hTx , y i

⇔ Tx = 0 i.e. x ∈ ker T .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 13

Kernel and image

ker T is stable through T . Observation

If T = T ^∗ , H = ker T ⊕ Im T , i.e. ker T = Im T ^⊥ i.e. (ker T ) ^⊥ = Im T . Proof.

x ∈ Im T ^⊥ means ∀y ∈ H

0 = hx , Ty i = hTx , y i

⇔ Tx = 0 i.e. x ∈ ker T .

Kernel and image

ker T is stable through T . Observation

If T = T ^∗ , H = ker T ⊕ Im T , i.e. ker T = Im T ^⊥ i.e. (ker T ) ^⊥ = Im T . Proof.

x ∈ Im T ^⊥ means ∀y ∈ H

0 = hx , Ty i = hTx , y i

⇔ Tx = 0 i.e. x ∈ ker T .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 5 / 13

Reduction to one-to-one operators

π ₀ , ⊥-proj on ker T , I − π ₀ ⊥-proj on Im T then (with previous observation)

T = π ₀ T π ₀ + (I − π ₀ )T (I − π ₀ ) both are self adjoint and compact if T is.

(I − π ₀ )T (I − π ₀ ) is T : Im T → Im T and is one-to-one.

Reduction to one-to-one operators

π ₀ , ⊥-proj on ker T , I − π ₀ ⊥-proj on Im T then (with previous observation)

T = π ₀ T π ₀ + (I − π ₀ )T (I − π ₀ ) both are self adjoint and compact if T is.

(I − π ₀ )T (I − π ₀ ) is T : Im T → Im T and is one-to-one.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 13

Reduction to one-to-one operators

π ₀ , ⊥-proj on ker T , I − π ₀ ⊥-proj on Im T then (with previous observation)

T = π ₀ T π ₀ + (I − π ₀ )T (I − π ₀ ) both are self adjoint and compact if T is.

(I − π ₀ )T (I − π ₀ ) is T : Im T → Im T and is one-to-one.

Reduction to one-to-one operators

π ₀ , ⊥-proj on ker T , I − π ₀ ⊥-proj on Im T then (with previous observation)

T = π ₀ T π ₀ + (I − π ₀ )T (I − π ₀ ) both are self adjoint and compact if T is.

(I − π ₀ )T (I − π ₀ ) is T : Im T → Im T and is one-to-one.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 13

Reduction to one-to-one operators

π ₀ , ⊥-proj on ker T , I − π ₀ ⊥-proj on Im T then (with previous observation)

T = π ₀ T π ₀ + (I − π ₀ )T (I − π ₀ ) both are self adjoint and compact if T is.

(I − π ₀ )T (I − π ₀ ) is T : Im T → Im T and is one-to-one.

Reduction to one-to-one operators

π ₀ , ⊥-proj on ker T , I − π ₀ ⊥-proj on Im T then (with previous observation)

T = π ₀ T π ₀ + (I − π ₀ )T (I − π ₀ ) both are self adjoint and compact if T is.

(I − π ₀ )T (I − π ₀ ) is T : Im T → Im T and is one-to-one.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 6 / 13

Spectral theorem - comments

one can also interpret this as T having a matrix of the form

ker T E 1 E 2 · · ·

e _1,1 , . . . , e _n

_,1 e _1,2 , . . . , e _n

_,2

ker T 0 0 0 · · ·

e _1,1 .. . e _n

_,1

0

λ ₁ 0

. ..

0 λ ₁

0 e _1,2

.. . e _n

_,2

0

λ ₂ 0

. ..

0 λ ₂

0 0

. .. 0

. ..

0 . ..

Eigenvalues are real

Observation

If T = T ^∗ , and λ eigenvalue: ∃x ∈ H, x 6= 0 s.t. Tx = λx then λ ∈ R . Proof.

λkx k ² = hλx , x i = hTx , x i = hx, Txi

= hx, λx i = ¯ λkxk ² .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 13

Eigenvalues are real

Observation

If T = T ^∗ , and λ eigenvalue: ∃x ∈ H, x 6= 0 s.t. Tx = λx then λ ∈ R . Proof.

λkx k ² = hλx , x i = hTx , x i = hx, Txi

= hx, λx i = ¯ λkxk ² .

Eigenvalues are real

Observation

If T = T ^∗ , and λ eigenvalue: ∃x ∈ H, x 6= 0 s.t. Tx = λx then λ ∈ R . Proof.

λkx k ² = hλx , x i = hTx , x i = hx, Txi

= hx, λx i = ¯ λkxk ² .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 13

Eigenvalues are real

Observation

If T = T ^∗ , and λ eigenvalue: ∃x ∈ H, x 6= 0 s.t. Tx = λx then λ ∈ R . Proof.

λkx k ² = hλx , x i = hTx , x i = hx, Txi

= hx, λx i = ¯ λkxk ² .

Eigenvalues are real

Observation

If T = T ^∗ , and λ eigenvalue: ∃x ∈ H, x 6= 0 s.t. Tx = λx then λ ∈ R . Proof.

λkx k ² = hλx , x i = hTx , x i = hx, Txi

= hx, λx i = ¯ λkxk ² .

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 8 / 13

Eigenvalues are real

Observation

If T = T ^∗ , and λ eigenvalue: ∃x ∈ H, x 6= 0 s.t. Tx = λx then λ ∈ R . Proof.

λkx k ² = hλx , x i = hTx , x i = hx, Txi

= hx, λx i = ¯ λkxk ² .

Eigenspaces are orthogonal

Observation

If T = T ^∗ , λ 6= µ, E _λ = ker(T − λI), E _µ corresponding eigenspaces then E _λ ⊥ E _µ

Proof.

x ∈ E _λ , y ∈ E µ

λhx , y i = hλx , y i = hTx , y i = hx, Ty i

= hx , µy i = ¯ µhx , y i = µhx, y i

E µ ⊂ E _λ ^⊥ and E µ =eigenspace for eigenvalue µ of (I − π _λ )T (I − π _λ ) i.e.

of T restricted to E _λ ^⊥

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 13

Eigenspaces are orthogonal

Observation

If T = T ^∗ , λ 6= µ, E _λ = ker(T − λI), E _µ corresponding eigenspaces then E _λ ⊥ E _µ

Proof.

x ∈ E _λ , y ∈ E µ

λhx , y i = hλx , y i = hTx , y i = hx, Ty i

= hx , µy i = ¯ µhx , y i = µhx, y i

E µ ⊂ E _λ ^⊥ and E µ =eigenspace for eigenvalue µ of (I − π _λ )T (I − π _λ ) i.e.

of T restricted to E _λ ^⊥

Eigenspaces are orthogonal

Observation

If T = T ^∗ , λ 6= µ, E _λ = ker(T − λI), E _µ corresponding eigenspaces then E _λ ⊥ E _µ

Proof.

x ∈ E _λ , y ∈ E µ

λhx , y i = hλx , y i = hTx , y i = hx, Ty i

= hx , µy i = ¯ µhx , y i = µhx, y i

E µ ⊂ E _λ ^⊥ and E µ =eigenspace for eigenvalue µ of (I − π _λ )T (I − π _λ ) i.e.

of T restricted to E _λ ^⊥

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 13

Eigenspaces are orthogonal

Observation

If T = T ^∗ , λ 6= µ, E _λ = ker(T − λI), E _µ corresponding eigenspaces then E _λ ⊥ E _µ

Proof.

x ∈ E _λ , y ∈ E µ

λhx , y i = hλx , y i = hTx , y i = hx, Ty i

= hx , µy i = ¯ µhx , y i = µhx, y i

E µ ⊂ E _λ ^⊥ and E µ =eigenspace for eigenvalue µ of (I − π _λ )T (I − π _λ ) i.e.

of T restricted to E _λ ^⊥

Eigenspaces are orthogonal

Observation

If T = T ^∗ , λ 6= µ, E _λ = ker(T − λI), E _µ corresponding eigenspaces then E _λ ⊥ E _µ

Proof.

x ∈ E _λ , y ∈ E µ

λhx , y i = hλx , y i = hTx , y i = hx, Ty i

= hx , µy i = ¯ µhx , y i = µhx, y i

E µ ⊂ E _λ ^⊥ and E µ =eigenspace for eigenvalue µ of (I − π _λ )T (I − π _λ ) i.e.

of T restricted to E _λ ^⊥

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 9 / 13

Eigenspaces are orthogonal

Observation

If T = T ^∗ , λ 6= µ, E _λ = ker(T − λI), E _µ corresponding eigenspaces then E _λ ⊥ E _µ

Proof.

x ∈ E _λ , y ∈ E µ

λhx , y i = hλx , y i = hTx , y i = hx, Ty i

= hx , µy i = ¯ µhx , y i = µhx, y i

E µ ⊂ E _λ ^⊥ and E µ =eigenspace for eigenvalue µ of (I − π _λ )T (I − π _λ ) i.e.

of T restricted to E _λ ^⊥

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 13

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 13

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 13

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 13

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 10 / 13

Eigenvalues exist

Lemma

If T 6= 0 is compact, self-adjoint the either kT k or −kT k is an eigenvalue

Proof: α := kT k then

α ² = sup{kTxk ² : kxk = 1} = sup{hTx , Tx i : kxk = 1}

= sup{ D T ² x , x

E

: kx k = 1}.

∃x _n , kx _n k = 1,

T ² x n , x n

→ α ² .

T ² x _n − α ² x _n

2

= T ² x _n

2 − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

≤ kT k ⁴ kx _n k ² − 2α ² < D

T ² x _n , x _n E

+ α ⁴ kx _n k ⁴

= 2α ⁴ − 2α ² < D

T ² x n , x n

E → 0.

Eigenvalues exist - proof continued

Proof continued.

T compact ⇒ T ² compact, x _n ∈ B,

T ² x _n − α ² x _n

2 → 0

Go to subsequence (T ² x n

) converges x n

also. x the limit, kx k = 1 and

T ² x − α ² x

= 0 i.e. (T ² − α ² )x = 0.

Rewrite this as (T + αI)(T − αI)x = 0

Case 1: (T − αI)x = 0 that is Tx = αx , α is an eigenvalue.

Case 2: y := (T − αI)x 6= 0 then (T + αI)y = 0 i.e. Ty = −αy , −α is an eigenvalue;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 11 / 13

Eigenvalues exist - proof continued

Proof continued.

T compact ⇒ T ² compact, x _n ∈ B,

T ² x _n − α ² x _n

2 → 0

Go to subsequence (T ² x n

) converges x n

also. x the limit, kx k = 1 and

T ² x − α ² x

= 0 i.e. (T ² − α ² )x = 0.

Rewrite this as (T + αI)(T − αI)x = 0

Case 1: (T − αI)x = 0 that is Tx = αx , α is an eigenvalue.

Case 2: y := (T − αI)x 6= 0 then (T + αI)y = 0 i.e. Ty = −αy , −α is

an eigenvalue;

Eigenvalues exist - proof continued

Proof continued.

T compact ⇒ T ² compact, x _n ∈ B,

T ² x _n − α ² x _n

2 → 0

Go to subsequence (T ² x n

) converges x n

also. x the limit, kx k = 1 and

T ² x − α ² x

= 0 i.e. (T ² − α ² )x = 0.

Rewrite this as (T + αI)(T − αI)x = 0

Case 1: (T − αI)x = 0 that is Tx = αx , α is an eigenvalue.

Case 2: y := (T − αI)x 6= 0 then (T + αI)y = 0 i.e. Ty = −αy , −α is an eigenvalue;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 11 / 13

Eigenvalues exist - proof continued

Proof continued.

T compact ⇒ T ² compact, x _n ∈ B,

T ² x _n − α ² x _n

2 → 0

Go to subsequence (T ² x n

) converges x n

also. x the limit, kx k = 1 and

T ² x − α ² x

= 0 i.e. (T ² − α ² )x = 0.

Rewrite this as (T + αI)(T − αI)x = 0

Case 1: (T − αI)x = 0 that is Tx = αx , α is an eigenvalue.

Case 2: y := (T − αI)x 6= 0 then (T + αI)y = 0 i.e. Ty = −αy , −α is

an eigenvalue;

Eigenvalues exist - proof continued

Proof continued.

T compact ⇒ T ² compact, x _n ∈ B,

T ² x _n − α ² x _n

2 → 0

Go to subsequence (T ² x n

) converges x n

also. x the limit, kx k = 1 and

T ² x − α ² x

= 0 i.e. (T ² − α ² )x = 0.

Rewrite this as (T + αI)(T − αI)x = 0

Case 1: (T − αI)x = 0 that is Tx = αx , α is an eigenvalue.

Case 2: y := (T − αI)x 6= 0 then (T + αI)y = 0 i.e. Ty = −αy , −α is an eigenvalue;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 11 / 13

Eigenvalues exist - proof continued

Proof continued.

T compact ⇒ T ² compact, x _n ∈ B,

T ² x _n − α ² x _n

2 → 0

Go to subsequence (T ² x n

) converges x n

also. x the limit, kx k = 1 and

T ² x − α ² x

= 0 i.e. (T ² − α ² )x = 0.

Rewrite this as (T + αI)(T − αI)x = 0

Case 1: (T − αI)x = 0 that is Tx = αx , α is an eigenvalue.

Case 2: y := (T − αI)x 6= 0 then (T + αI)y = 0 i.e. Ty = −αy , −α is

an eigenvalue;

Eigenvalues exist - proof continued

Proof continued.

T compact ⇒ T ² compact, x _n ∈ B,

T ² x _n − α ² x _n

2 → 0

Go to subsequence (T ² x n

) converges x n

also. x the limit, kx k = 1 and

T ² x − α ² x

= 0 i.e. (T ² − α ² )x = 0.

Rewrite this as (T + αI)(T − αI)x = 0

Case 1: (T − αI)x = 0 that is Tx = αx , α is an eigenvalue.

Case 2: y := (T − αI)x 6= 0 then (T + αI)y = 0 i.e. Ty = −αy , −α is an eigenvalue;

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 11 / 13

Eigenvalues exist - proof continued

Proof continued.

T compact ⇒ T ² compact, x _n ∈ B,

T ² x _n − α ² x _n

2 → 0

Go to subsequence (T ² x n

) converges x n

also. x the limit, kx k = 1 and

T ² x − α ² x

= 0 i.e. (T ² − α ² )x = 0.

Rewrite this as (T + αI)(T − αI)x = 0

Case 1: (T − αI)x = 0 that is Tx = αx , α is an eigenvalue.

Case 2: y := (T − αI)x 6= 0 then (T + αI)y = 0 i.e. Ty = −αy , −α is

an eigenvalue;

Eigenspaces have finite dimension

Observation

If T is compact, each eigenspace E _λ (λ 6= 0) has finite dimension.

Proof.

(e _k ) _k∈I O.N.B of E _λ , Te _k = λe _k . If k 6= `, kTe <sub>k</sub> − Te <sub>`</sub> k = |λ| √ 2 6= 0.

Can not extract convergent subsequence (if I infinite) Contradicts T compact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 12 / 13

Eigenspaces have finite dimension

Observation

If T is compact, each eigenspace E _λ (λ 6= 0) has finite dimension.

Proof.

(e _k ) _k∈I O.N.B of E _λ , Te _k = λe _k . If k 6= `, kTe <sub>k</sub> − Te <sub>`</sub> k = |λ| √ 2 6= 0.

Can not extract convergent subsequence (if I infinite)

Contradicts T compact.

Eigenspaces have finite dimension

Observation

If T is compact, each eigenspace E _λ (λ 6= 0) has finite dimension.

Proof.

(e _k ) _k∈I O.N.B of E _λ , Te _k = λe _k . If k 6= `, kTe <sub>k</sub> − Te <sub>`</sub> k = |λ| √ 2 6= 0.

Can not extract convergent subsequence (if I infinite) Contradicts T compact.

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 12 / 13

Eigenspaces have finite dimension

Observation

If T is compact, each eigenspace E _λ (λ 6= 0) has finite dimension.

Proof.

(e _k ) _k∈I O.N.B of E _λ , Te _k = λe _k . If k 6= `, kTe <sub>k</sub> − Te <sub>`</sub> k = |λ| √ 2 6= 0.

Can not extract convergent subsequence (if I infinite)

Contradicts T compact.

That’s all!

Thank you for your attention!

Next video: The proof.

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

Philippe Jaming (Universit ´e de Bordeaux) Spectral Theorem 2 Master Math & Applications 13 / 13

That’s all!

Thank you for your attention!

Next video: The proof.

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