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Let $sigma(A)$, $rho(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$rho(AB)le r(A)r(B) quadtext{ for all bounded linear operators } B$$ if and only if there is a unique $mu in sigma (A)$ satisfying $|mu| = rho(A)$ and $A = frac{mu(I + L)}{2}$ for a contraction $L$ with $1insigma(L)$. One can get the same conclusion on $A$ if $rho(AB) le r(A)r(B)$ for all rank one operators $B$. If $H$ is of finite dimension, we can further decompose $L$ as a direct sum of $C oplus 0$ under a suitable choice of orthonormal basis so that $Re(C^{-1}x,x) ge 1$ for all unit vector $x$.
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We examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the $n^2$ by $n^2$ matrix $sum{overline{X_i}}otimes{X_i}.$ We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely postive map, including the special case of quantum channels. The average of the iterates of a completely positive map approach to a completely positive map where the Kraus operators span an ideal in the algebra generated by the Kraus operators of the original completely positive map. We also give an elementary treatment of Popescus theorems on similarity to row contractions in the matrix case, describe connections to the Parrilo-Jadbabaie relaxation, and give a detailed analysis of the maximal spectrum of a completely positive map.
Let k, p, q be positive integers with k < p < q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph Kp,q of bipartition orders p and q by deleting k edges is attained when the deleting edges are all incident on a common vertex which is located in the partite set of order q. Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences.
A Banach space operator $Tin B(X)$ is left polaroid if for each $lambdainhbox{iso}sigma_a(T)$ there is an integer $d(lambda)$ such that asc $(T-lambda)=d(lambda)<infty$ and $(T-lambda)^{d(lambda)+1}X$ is closed; $T$ is finitely left polaroid if asc $(T-lambda)<infty$, $(T-lambda)X$ is closed and $dim(T-lambda)^{-1}(0)<infty$ at each $lambdainhbox{iso }sigma_a(T)$. The left polaroid property transfers from $A$ and $B$ to their tensor product $Aotimes B$, hence also from $A$ and $B^*$ to the left-right multiplication operator $tau_{AB}$, for Hilbert space operators; an additional condition is required for Banach space operators. The finitely left polaroid property transfers from $A$ and $B$ to their tensor product $Aotimes B$ if and only if $0 otinhbox{iso}sigma_a(Aotimes B)$; a similar result holds for $tau_{AB}$ for finitely left polaroid $A$ and $B^*$.
Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give a sharp upper bound for rho(G) by a function of d_1, d_2, ..., d_L, which generalizes a series of previous results.
In this paper, we give some bounds for principal eigenvector and spectral radius of connected uniform hypergraphs in terms of vertex degrees, the diameter, and the number of vertices and edges.
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