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Math Help - Unitary square root

  1. #1
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    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?

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  2. #2
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    Re: Unitary square root

    Quote Originally Posted by H12504106 View Post
    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.
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