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Give an example of a normal operator $T$ on a complex inner product space, which is an isometry but $T^2≠I_V$.

(This question did not give what the inner product is, so how should I do? If under dot product, does T=(0 1; -1 0) satisfy?

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    I think the question suggests you are allowed to choose a complex inner product space of you liking, and define an operator on that. The answer you suggested (if that is a $2\times2$ matrix you intended to write down) supposes the space $\Bbb C^2$. However there are also answers that can be defined with the same formualtion on _any_ nonzero complex inner product space.2012-12-02
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    Marc van Leeuwen: Can you give an example for on any nonzero complex inner product space?2012-12-02
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    Multiplication by the scalar $i$2012-12-02
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    May I suggest that instead of flooding this website with vast numbers of related questions, that you STOP and wait and digest the answers you get and see whether you can apply them to the rest of your questions?2012-12-02
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    This seems like a strange question. The condition $T^2 = I$ is strong - most isometries should _not_ obey it. Just curious - did you forget a part of the question?2012-12-02
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    Marc van Leeuwen: Do you mean (0 i; -i 0)? Why this satisfies all inner products?2012-12-02

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