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In this paper we study the spectrum $Sigma$ of the infinite Feinberg-Zee random hopping matrix, a tridiagonal matrix with zeros on the main diagonal and random $pm 1$s on the first sub- and super-diagonals; the study of this non-selfadjoint random matrix was initiated in Feinberg and Zee (Phys. Rev. E 59 (1999), 6433--6443). Recently Hagger (arXiv:1412.1937, Random Matrices: Theory Appl.}, {bf 4} 1550016 (2015)) has shown that the so-called periodic part $Sigma_pi$ of $Sigma$, conjectured to be the whole of $Sigma$ and known to include the unit disk, satisfies $p^{-1}(Sigma_pi) subset Sigma_pi$ for an infinite class $S$ of monic polynomials $p$. In this paper we make very explicit the membership of $S$, in particular showing that it includes $P_m(lambda) = lambda U_{m-1}(lambda/2)$, for $mgeq 2$, where $U_n(x)$ is the Chebychev polynomial of the second kind of degree $n$. We also explore implications of these inverse polynomial mappings, for example showing that $Sigma_pi$ is the closure of its interior, and contains the filled Julia sets of infinitely many $pin S$, including those of $P_m$, this partially answering a conjecture of the second author.
This paper provides a new proof of a theorem of Chandler-Wilde, Chonchaiya and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more g
For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $mathfrak{H}$ singular perturbations of the form $A_0+V, V=sum_{1}^{n}{b}_{ij}<psi_j,cdot>psi_i$ are studied under some additional requirements of symmetry imposed on the initia
This short note presents upper bounds of the expectations of the largest singular values/eigenvalues of various types of random tensors in the non-asymptotic sense. For a standard Gaussian tensor of size $n_1timescdotstimes n_d$, it is shown that the
Large-dimensional random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large dimensional systems. With the fact that we have entered an unpre
We analyze spectral properties of the Hilbert $L$-matrix $$left(frac{1}{max(m,n)+ u}right)_{m,n=0}^{infty}$$ regarded as an operator $L_{ u}$ acting on $ell^{2}(mathbb{N}_{0})$, for $ uinmathbb{R}$, $ u eq0,-1,-2,dots$. The approach is based on a spe