Question

) Let T be a normal operator on a finite dimensional Hilbert space H

with spectrum𝜆1, 𝜆2, … … , 𝜆𝑚 , . Then prove that

i) T is self - adjoint⟺ each 𝜆𝑖,,is real

ii) T is positive ⟺ each 𝜆𝑖 ≥ 0

iii) T is unitary ⟺ 𝜆𝑖 =1 for each .

Answer 1

A complex number λ ∈ C is called an eigenvalue of T ∈ B(H) if there exists a vector 0 $\neq$ x ∈ H such that Tx = λx

i)

Let λ be an eigenvalue of T and x be the corresponding eigenvector. Then Tx = λx.

$\lambda||x||^ 2 = λ \langle x,x\rangle = \langle λ x,x\rangle = \langle T x,x\rangle = \langle x,Tx\rangle= \langle x, λ x\rangle =\overline{\lambda}||x||^ 2$

Since x $\neq$ 0, we have λ = $\overline{ λ}$

ii)

T is positive if $Tx\ge 0$

then Tx = λx $\ge 0 \implies \lambda \ge 0$

iii)

$Tx=T^{-1}x$

$\lambda x=\lambda^{-1} x \implies \lambda=1$