The spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its spectrum.

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### Finding limit of spectral radius to size ratios on a sequence of symmetric matrixes increasing in dimensions

I came across this problem recently: let $r_n$ be the spectral radius of the $n\times n$ matrix $A_n$ defined by ${A_n}_{(i,j)}=n-|i-j|$, find $$\lim_{n\to\infty}\frac{r_n}{n^2}$$ Intuitively, I guess ...
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### How does the spectral radius of the Hessian of a box-constrained polynomial with random coefficients change with degree and dimension?

It seems that spectral radius of a symmetric random matrix with i.i.d. entries 0 mean and some assumptions on variance converges to $\sqrt(n)$ where $n$ is the number of dimensions: e.g. https://arxiv....
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### Proof that the hausdorff distance $d_H( \sigma (A), \sigma(B)) \le r(A-B)$ for commuting operators $A,B$

I am supposed to prove that for two bounded commuting operators $A, B$ on a banach space, the hausdorff distance of the spectra $d_H(\sigma(A),\sigma(B))$ is less than or equal to the spectral radius ...
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### Smallest $k$ such that $|\langle ABx,x\rangle| \le k\langle Ax,x\rangle$, where $A\ge 0$ and $AB$ self-adjoint

Let $A$ and $B$ be linear transformetions of a finite-dimensional inner product space $V$. Let $A$ be positive semi-definite and assume that $AB$ is self-adjoint. Then what is the smallest $k\ge0$ ...
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### Showing matrix C has spectral radius less than 1

We are given that $A$ is symmetric and positive definite, and $B$ is such that $A-B-B^T$ is also symmetric and positive definite. We are asked to show that $C=-(A-B)^{-1} B$ has spectral radius less ...
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### Spectral Radius at the Unique and Globally Attractive Fixed Point of a Specific Type of Mapping

Consider a mapping $f: \mathbb{R}_+^n \rightarrow \mathbb{R}_{++}^n$. If it is monotonic and strictly subhomogeous, then it is contractive under the Thompson’s metric (See Lemma 2.1.7). Here, ...
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I have a question concerning exercise 6 in chapter 2 of Murphy's $C^*$-algebras and Operator Theory. The exercise is as follows: Let A be a unital $C^*$-algebra. If $r(a) < 1$ and $b = (\sum_{n=0}... • 918 4 votes 1 answer 208 views ### Character Space of$C^1[0,1]\$ and Gelfand Representation

I have recently been working through some exercises in Murphy's "C*-Algebras and Operator Theory," and I am having some trouble with Exercise 10 in Chapter 1. The exercise is as follows: Let ...
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