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How do I prove these two theorems? Furthermore, can I apply them to infinite-dimensional spaces, such as Banach spaces?

Theorem 1. Let $M\in \mathbb{C}_{n\times n}$ be a matrix and $\epsilon > 0$ be given. There is at least one matrix norm $||\cdot||$ such that $$\rho(M) \leq ||M|| \leq \rho(M) + \epsilon$$ where $\rho(M) = \max\{|\lambda_1(M)|, \dots , |\lambda_n(M)|\}$ denotes the spectral radius of $M$.

Theorem 2. If $P \in \mathbb{C}_{n\times n}$ and $S\in \mathbb{C}_{n\times n}$ are such that $P = P^2$ and $PS = SP$ then $$\rho(PS) \leq \rho(S).$$

I have used these results in finite dimensional spaces and want to use them in a Banach space.

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    I'm puzzled about your Theorem 1. Is it for a fixed $M$? In that case, wouldn't the result be trivial, by just scaling appropriately some norm?2012-05-07
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    It's general result. My main concern is can we apply these theorems in Banach space as well?2012-05-07
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    @MartinArgerami: I think "matrix norm" here includes submultiplicativity, which is why you can't just rescale any norm.2012-05-08
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    Can't i apply these theprems on Banach space operators?2012-05-08
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    @Jonas: I know now, I searched the term on Horn and Johnson when looking for the proof of Theorem 1.2012-05-08
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    @jonas Can you explain sir?2012-05-08

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