Spectral Theorem for compact self-adjoint operators

(1)

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 Master Math & Applications 1 / 9

(2)

Preliminary warning

This video is a complement to the lecture notes available at

http://www.u-bordeaux.fr/˜ pjaming/enseignement/M1.html I also assume that you have followed previous courses in which the following notions were presented

– Self-adjoint operators on Hilbert spaces :hTx,yi=hx,Tyifor every x,y

– compact operators :T(B_H)is compact (B_H unit ball ofH)

or equivalently, for every bounded sequence(xn)n∈N⊂H, there exists a subsequence of(Tx_n)that converges inH(stongly)

or equivalently, if the sequence(x_n)n∈N∈His weakly convergent, then (Txn)n∈Nis strongly convergent.

(3)

Preliminary warning

(4)

Preliminary warning

(5)

Preliminary warning

(6)

Preliminary warning

(7)

Preliminary warning

(8)

Spectral Theorem for compact self-adjoint operators

Theorem (Spectral Theorem)

H Hilbert space.T op ´erator H →Hcompact,self-adjoint. Then

∃a set I finite or countable

∃a decomposition of H into mutually orthogonal subspaces H = kerT ⊕M

i∈I

E_i = kerT ⊕rangeT

where each E_i hasfinite dimension;

∃arealsequence(λ_i)_i∈I with|λ_i|non-increasing and if I infinite, λ_i →0;

s.t.

T =X

i∈I

λ_iπ_i π_i orthogonal projection on E_i.

(9)

Spectral Theorem for compact self-adjoint operators

H Hilbert space. T op ´erator H →Hcompact,self-adjoint. Then

i∈I

s.t.

T =X

i∈I

(10)

Spectral Theorem for compact self-adjoint operators

i∈I

s.t.

T =X

i∈I

(11)

Spectral Theorem for compact self-adjoint operators

H Hilbert space. T op ´erator H →Hcompact,self-adjoint.Then

i∈I

s.t.

T =X

i∈I

(12)

Spectral Theorem for compact self-adjoint operators

i∈I

s.t.

T =X

i∈I

(13)

Spectral Theorem for compact self-adjoint operators

i∈I

s.t.

T =X

i∈I

(14)

Spectral Theorem for compact self-adjoint operators

i∈I

s.t.

T =X

i∈I

(15)

Spectral Theorem for compact self-adjoint operators

i∈I

s.t.

T =X

i∈I

(16)

Spectral Theorem for compact self-adjoint operators

i∈I

∃arealsequence(λ_i)_i∈I with|λ_i|non-increasingand if I infinite, λ_i →0;

s.t.

T =X

i∈I

(17)

Spectral Theorem for compact self-adjoint operators

i∈I

s.t.

T =X

i∈I

(18)

Spectral theorem - comments

Choose(e_k,i)_k=1,...,n_i (n_i = dimE_i) ono.n.b.ofE_i,

π_ix =

n_i

X

k=1

x,e_k_,i e_k,i

thus

Tx =X

i∈I n_i

X

k=1

λ_i x,e_k_,i

e_k,i.

That is,T isdiagonalisablein an orthonormal basis of eigenvectors AttentionTo obtain a basis ofH, we mut complete with a basis ofkerT which requireskerT to beseparable(thusH as well).

(19)

Spectral theorem - comments

π_ix =

n_i

X

k=1

x,e_k_,i e_k,i

thus

Tx =X

i∈I n_i

X

k=1

λ_i x,e_k_,i

e_k,i.

(20)

Spectral theorem - comments

π_ix =

n_i

X

k=1

x,e_k_,i e_k,i

thus

Tx =X

i∈I n_i

X

k=1

λ_i x,e_k_,i

e_k,i.

(21)

Spectral theorem - comments

π_ix =

n_i

X

k=1

x,e_k_,i e_k,i

thus

Tx =X

i∈I n_i

X

k=1

λ_i x,e_k_,i

e_k,i.

(22)

Spectral theorem - comments

π_ix =

n_i

X

k=1

x,e_k_,i e_k,i

thus

Tx =X

i∈I n_i

X

k=1

λ_i x,e_k_,i

e_k,i.

(23)

Spectral theorem - comments

We would then like to take,|λ_k|decreasing to 0 and(e_k)_k∈_No.n.b. ofH and

Tx =X

k∈N

λ_khx,e_kie_k

To do so, rewrite the basis[

i∈I

(e_k,i)_k=1,...,n_i and add a basis ofkerT. IfkerT has finite dimension, put this basis at the beginning if its co-dimension is finite, put it at the end.

Otherwise we have to mix the 2 bases and lose the fact that|λ_k|is non-increasing

Let us put it in practice

(24)

Spectral theorem - comments

Tx =X

k∈N

λ_khx,e_kie_k

i∈I

(25)

Spectral theorem - comments

Tx =X

k∈N

λ_khx,e_kie_k

i∈I

(e_k,i)_k=1,...,n_i and add a basis ofkerT. IfkerT has finite dimension, put this basis at the beginningif its co-dimension is finite, put it at the end.

(26)

Spectral theorem - comments

Tx =X

k∈N

λ_khx,e_kie_k

i∈I

(27)

Spectral theorem - comments

Tx =X

k∈N

λ_khx,e_kie_k

i∈I

(28)

Spectral theorem - comments

Tx =X

k∈N

λ_khx,e_kie_k

i∈I

(29)

Th ´eor `eme spectral - comments

e_1,1, . . . ,e_n₁_,1basis ofE₁→e˜₁, . . . ,e˜_n₁ e_1,2, . . . ,e_n₁_,2basis ofE₂→e˜_n₁₊₁, . . . ,e˜n₁+n₂

either it stops at somee˜_N (caseco−dim kerT <+∞) or gives an infinite sequence

take a basis ofkerT(f₁, . . . ,f_M)ifM = dim kerT <+∞or(f_k)k∈N

otherwise.

(30)

Th ´eor `eme spectral - comments

otherwise.

(31)

Th ´eor `eme spectral - comments

otherwise.

(32)

Th ´eor `eme spectral - comments

otherwise.

(33)

Th ´eor `eme spectral - comments

otherwise.

(34)

Th ´eor `eme spectral - comments

otherwise.

(35)

Th ´eor `eme spectral - comments

otherwise.

(36)

Th ´eor `eme spectral - comments

otherwise.

(37)

Th ´eor `eme spectral - comments

otherwise.

(38)

Spectral theorem - comments

Unite both bases

Ifco−dim kerT <+∞,(e_k)k≥1is→˜e₁, . . . ,e˜_N,f₁, . . .the

corresponding eigenvalues areλ₁, . . . , λ_N,0. . .are decreasing to 0 Ifdim kerT <+∞,(e_k)_k_≥1is→f₁, . . . ,f_M,˜e₁, . . .the corresponding eigenvalues are 0, . . . ,0, λ₁, . . .are not decreasing but go to 0 Ifdim kerT =co−dim kerT = +∞,(e_k)k≥1is→e˜₁,f₁,e˜₂,f₂, . . .the corresponding eigenvalues areλ₁,0, λ₂,0, . . .are not decreasing but go to 0.

Theorem (Spectral Theorem revisited)

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.

(39)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(40)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(41)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(42)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(43)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(44)

Spectral theorem - comments

Unite both bases

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

Tx =X

k∈N

λ_khx,e_kie_k.

(45)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(46)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(47)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(48)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(49)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(50)

Spectral theorem - comments

Unite both bases

Tx =X

k∈N

λ_khx,e_kie_k.

(51)

Spectral theorem - Finite dimensional case

Finite dimensional case : self adjoint matrices are diagonalizable in an orthonormal basis of eigenvectorsA=A^∗ ⇒∃U unitary (U⁻¹=U^∗)∃D real diagonalA=UDU^∗.

(52)

Spectral theorem - Finite dimensional case

(53)

Spectral theorem - Finite dimensional case

(54)

Spectral theorem - Finite dimensional case

(55)

Spectral theorem - Finite dimensional case

(56)

Spectral theorem - comments

one can also interpret this asT having 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 ... e_n₁_,1

0

λ₁ 0

. ..

0 λ₁

0 e_1,2

... e_n₂_,2

0

λ₂ 0

. ..

0 λ₂

0 0

. .. 0

. ..

0 . ..

(57)

That’s all !

Thank you for your attention ! Next video : an example.

(58)

That’s all !

Thank you for your attention ! Next video : an example.