Kenneth Kuttler

300 CHAPTER 12. SELF ADJOINT OPERATORS

where C is the upper triangular matrix which has cij for i ≤ j and zeros elsewhere. Thisequals 0 if and only if λ is one of the diagonal entries, one of the ckk. ■

Now with the above Schur’s theorem, the following diagonalization theorem comes veryeasily. Recall the following definition.

Definition 12.2.3 Let L ∈ L (H,H) where H is a finite dimensional inner product space.Then L is Hermitian if L∗ = L.

Theorem 12.2.4 Let L ∈ L (H,H) where H is an n dimensional inner product space. IfL is Hermitian, then all of its eigenvalues λk are real and there exists an orthonormal basisof eigenvectors {wk} such that

L =∑k

λkwk⊗wk.

Proof: By Schur’s theorem, Theorem 12.2.2, there exist lij ∈ F such that

L =

n∑j=1

j∑i=1

lijwi⊗wj

Then by Lemma 11.4.2,

n∑j=1

j∑i=1

lijwi⊗wj = L = L∗ =

n∑j=1

j∑i=1

(lijwi⊗wj)∗

n∑j=1

j∑i=1

lijwj⊗wi =

n∑i=1

i∑j=1

ljiwi⊗wj

By independence, if i = j, lii = lii and so these are all real. If i < j, it follows fromindependence again that lij = 0 because the coefficients corresponding to i < j are all 0 onthe right side. Similarly if i > j, it follows lij = 0. Letting λk = lkk, this shows

L =∑k

λkwk ⊗wk

That each of these wk is an eigenvector corresponding to λk is obvious from the definitionof the tensor product. ■

12.3 Spectral Theory of Self Adjoint Operators

The following theorem is about the eigenvectors and eigenvalues of a self adjoint operator.Such operators are also called Hermitian as in the case of matrices. The proof given gen-eralizes to the situation of a compact self adjoint operator on a Hilbert space and leads tomany very useful results. It is also a very elementary proof because it does not use thefundamental theorem of algebra and it contains a way, very important in applications, offinding the eigenvalues. This proof depends more directly on the methods of analysis thanthe preceding material. Recall the following notation.

Definition 12.3.1 Let X be an inner product space and let S ⊆ X. Then

S⊥ ≡ {x ∈ X : (x, s) = 0 for all s ∈ S} .

Note that even if S is not a subspace, S⊥ is.

300 CHAPTER 12. SELF ADJOINT OPERATORSwhere C is the upper triangular matrix which has c;; for 1 < 7 and zeros elsewhere. Thisequals 0 if and only if A is one of the diagonal entries, one of the cy,. HfNow with the above Schur’s theorem, the following diagonalization theorem comes veryeasily. Recall the following definition.Definition 12.2.3 Let L © £(H,H) where H is a finite dimensional inner product space.Then L is Hermitian if L* = L.Theorem 12.2.4 Let L € £L(H,H) where H is an n dimensional inner product space. IfL is Hermitian, then all of its eigenvalues A, are real and there exists an orthonormal basisof eigenvectors {w;,} such thatL= S- ALWEOW.kProof: By Schur’s theorem, Theorem 12.2.2, there exist [,; € F such thatn oJL= S- S- lijwi@w;j=l i=lThen by Lemma 11.4.2,n 9 njLjwi@w; = L=L= S- (lijwi@w;)”j=l i=l j=l i=1n Jj __ not= 1; wj@wi = S- S- Lj.wi;Qw;j=l i=l i=1 j=1By independence, if i = j, 1, = 1; and so these are all real. If i < j, it follows fromindependence again that l,; = 0 because the coefficients corresponding to 7 < 7 are all 0 onthe right side. Similarly if 7 > 7, it follows 1;; = 0. Letting Ay = Ix, this showsL= So \pwe ® WekThat each of these wy is an eigenvector corresponding to Ax is obvious from the definitionof the tensor product.12.3. Spectral Theory of Self Adjoint OperatorsThe following theorem is about the eigenvectors and eigenvalues of a self adjoint operator.Such operators are also called Hermitian as in the case of matrices. The proof given gen-eralizes to the situation of a compact self adjoint operator on a Hilbert space and leads tomany very useful results. It is also a very elementary proof because it does not use thefundamental theorem of algebra and it contains a way, very important in applications, offinding the eigenvalues. This proof depends more directly on the methods of analysis thanthe preceding material. Recall the following notation.Definition 12.3.1 Let X be an inner product space and let S C X. ThenSt ={xeX:(a,s)=0 for alls € S}.Note that even if S is not a subspace, S+ is.