Proving two results about the spectral radius

Question

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.

Answer 1

Note that neither of your theorems make sense in a Banach space. The notion of "matrix norm" requires the norm to be submultiplicative, i.e. it assumes that you can multiply your elements. So does the definition of spectrum.

So a reasonable context to ask your questions is that of a Banach algebra.

I cannot really say anything about the first question. The proof in Horn and Johnson is very specific for matrices, but of course there might be another proof.

The second assertion does hold in any Banach algebra: since $\|PSP\|\leq\|P\|\,\|S\|\,\|P\|$ $$ \rho(PS)=\rho(PSP)=\lim_{n\to\infty}\|(PSP)^n\|^{1/n}=\lim_{n\to\infty}\|P^nS^nP^n\|^{1/n}\\\leq\lim_{n\to\infty}\|P^n\|^{1/n}\lim_{n\to\infty}\|S^n\|^{1/n}\lim_{n\to\infty}\|P^n\|^{1/n}=\rho(P)^2\,\rho(S). $$ But as $P^2=P$, any $\lambda$ in the spectrum of $P$ will satisfy $\lambda^2=\lambda$ (because $$P^2-\lambda^2 I=(P-\lambda I)(P+\lambda I),$$ so $P^2-\lambda^2 I$ is invertible if and only if $P-\lambda I$ is invertible). So $\rho(P)\leq1$.

Answer 2

Your Theorem 1 can be found in Horn and Johnson's ``Matrix Analysis".

For Theorem 2, $PS=SP$ implies the eigenvalues of $PS$ are of the form $\lambda(P)\lambda(S)$, where $\lambda(P), \lambda(S)$ are some eigenvalues of $P, S$, respectively. Now since $P$ is idempotent, its eigenvalues are $1$'s and $0$'s.

Answer 3

The first result is true in a unital C*-algebra, and in particular in the algebra of bounded operators on a Hilbert space. ($M_n$ can be viewed as the finite-dimensional case of this.) This follows from a result in an exercise in Murphy's book which was asked about recently here.

If $A$ is a unital C*-algebra, then for each $a\in A$ and each $\varepsilon>0$, there exists an invertible element $b$ of $A$ such that $\rho(a)\leq \|bab^{-1}\|<\rho(a)+\varepsilon$. We can define a new Banach algebra norm on $A$ by $|x|:=\|bxb^{-1}\|$.