Unitary operator on inner product space

Question

"Prove that if T is a unitary operator on a finite dimensional inner product space,then there exists a unitary operator U such that T^2=U" My question is,is it possible to solve the question even if the space is considered over the real field? Or is it possible only if the field is complex?

Answer 1

I assume you meant to say there exists a unitary $U$ such that $U^2 = T$.

If this is the case, then the statement is not true over the real numbers (or in general, if the base field fails to be algebraically closed). For example, there is no real unitary operator $U$ such that $$ U^2 = \pmatrix{1&0\\0&-1} $$