Question

Let (V,<,>) be an inner product space over C and T belongs to A(V) . Prove that if <Tx , Ty>=<x,y> for all x,y belongs to V , then T is unitary.

Answer 1

Answer on Question#40159 - Math – Linear Algebra:

Let $(V, <, >)$ be an inner product space over $\mathbb{C}$ and $T$ belongs to $(V)$. Prove that if

$$\forall x, y \in V: <tx, ty=""> = <x, y="">, \text{ then } T \text{ is unitary}.$$

Solution.

We need to prove that $T^*T = I$, where $T^*$ is an adjoin operator, and $I$ is an identity operator.

$$\begin{array}{l} \forall x, y \in V: (x, y) = (Tx, Ty) = (x, T^*Ty) \Rightarrow \forall x, y \in V: (x, T^*Ty - y) = 0 \Rightarrow \\ \Rightarrow \forall y \in V: T^*Ty - y = 0 \Rightarrow \forall y \in V: T^*Ty = y \Rightarrow T^*T = I. \end{array}$$