# Math Help - inner product

1. ## inner product

Can you help me with this. I don’t know what to do
Let R, S, T are linear operators, where V is a complex inner product space.
(i) Suppose that S is an isometry and R is a positive operator such that T=SR. Prove that R=square root of (T*T)
(ii) Let σ denote the smallest singular value of T, and let σ*denote the largest singular value of T. Prove that σ<=|| T(v)/||v|| ||<= σ* for every nonzero v in V.

Thanks

2. Originally Posted by bamby
Let R, S, T be linear operators, where V is a complex inner product space.
(i) Suppose that S is an isometry and R is a positive operator such that T=SR. Prove that R=square root of (T*T)
(ii) Let σ denote the smallest singular value of T, and let σ*denote the largest singular value of T. Prove that σ<=|| T(v)/||v|| ||<= σ* for every nonzero v in V.
(i) If S is an isometry then S*S = I. Therefore $T^*T = RS^*SR = R^2$. But a positive operator has a unique positive square root, so $R=(T^*T)^{1/2}$.

(ii) $\|Tv\|^2 = \langle T^*Tv,v\rangle = \langle R^2v,v\rangle$. But $\sigma^2 I\leqslant R^2\leqslant \sigma^{*2}I$. So $\sigma^2\langle v,v\rangle \leqslant\langle R^2v,v\rangle\leqslant \sigma^{*2}\langle v,v\rangle$, from which $\sigma\|v\|\leqslant\|Tv\|\leqslant\sigma^*\|v\|$.