Let

be a linear operator on a finite-dimensional unitary space

such that, for every

the following implication holds:

.

Prove that then there exists a scalar

and a unitary operator

(where

) such that

.

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I'm aware of the following facts:

if an operator

is unitary, and

, then for every

we have

Also, on a unitary space

the following is true for every

:

And finally, if

is a unitary operator, then if

is an orthonormal basis for

, then

is also an orthonormal basis for

.

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But so far, I've had no luck in employing these to solve this problem, so would greatly appreciate your help!