310 CHAPTER 12. SELF ADJOINT OPERATORS

such that

x =

r∑k=1

ckUxk +

n∑k=r+1

dkyk

Define

Rx ≡r∑

k=1

ckFxk +

n∑k=r+1

dkzk (12.13)

Thus

|Rx|2 =

r∑k=1

|ck|2 +n∑

k=r+1

|dk|2 = |x|2 .

Therefore, by Lemma 12.7.1 R∗R = I.Then also there exist unique scalars bk such that for a given x ∈ X,

Ux =r∑

k=1

bkUxk (12.14)

and so from 12.13,

RUx =

r∑k=1

bkFxk = F

(r∑

k=1

bkxk

)Is F (

∑rk=1 bkxk) = F (x)?(

F

(r∑

k=1

bkxk

)− F (x) , F

(r∑

k=1

bkxk

)− F (x)

)

=

((F ∗F )

(r∑

k=1

bkxk − x

),

(r∑

k=1

bkxk − x

))

=

(U2

(r∑

k=1

bkxk − x

),

(r∑

k=1

bkxk − x

))

=

(U

(r∑

k=1

bkxk − x

), U

(r∑

k=1

bkxk − x

))

=

(r∑

k=1

bkUxk − Ux,

r∑k=1

bkUxk − Ux

)= 0

Because from 12.14, Ux =∑r

k=1 bkUxk. Therefore, RUx = F (∑r

k=1 bkxk) = F (x). ■The following corollary follows as a simple consequence of this theorem. It is called the

left polar decomposition.

Corollary 12.7.3 Let F ∈ L (X,Y ) and suppose n ≥ m where X is a inner product space ofdimension n and Y is a inner product space of dimension m. Then there exists a HermitianU ∈ L (X,X) , and an element of L (X,Y ) , R, such that

F = UR, RR∗ = I.

Proof: Recall that L∗∗ = L and (ML)∗

= L∗M∗. Now apply Theorem 12.7.2 toF ∗ ∈ L (Y,X). Thus, F ∗ = R∗U where R∗ and U satisfy the conditions of that theorem.Then F = UR and RR∗ = R∗∗R∗ = I. ■

The following existence theorem for the polar decomposition of an element of L (X,X)is a corollary.