Thread: Unitary square root

Prove that if T is a unitary operator on a finite dimensional inner product space V, then T has a unitary square root. ie. there exists a unitary operator U such that T = U^2

I know that t is unitary if TT* = T*T = I, where T* is the adjoint of T
I am able to show that such a U is unitary (UUU*U* = I, therefore UU* = U*U) but i can show the existence of such a U. Anyone can help me to start the problem. Should I convert the operator to a matrix and work with matrix?

Thank You

Originally Posted by H12504106

Prove that if T is a unitary operator on a finite dimensional inner product space V, then T has a unitary square root. ie. there exists a unitary operator U such that T = U^2

I know that t is unitary if TT* = T*T = I, where T* is the adjoint of T
I am able to show that such a U is unitary (UUU*U* = I, therefore UU* = U*U) but i can show the existence of such a U. Anyone can help me to start the problem. Should I convert the operator to a matrix and work with matrix?

A unitary operator is normal, hence diagonalisable. So it can be represented by a diagonal matrix, all of whose diagonal elements are on the unit circle in the complex plane. Take U to be (the operator whose matrix is) the diagonal matrix with diagonal elements given by square roots of those of T.