An inequality from the “Interlacing-1” paper

This question is in reference to this paper, (or its arxiv version, For the argument to work one seems to need that for the symmetric ±1 signing adjacency matrix of the graph, As, it holds that, max−root(∑s∈{0,1}mdet(xI−As))≤max−root(Es∈{0,1}m[det(xI−As)]) But why should this inequality be true? and the argument works because the polynomial on the … Read more

quasi-nilpotent part of a dual operator

Definitions and notation. Let X be a complex Banach space and T∈L(X) a continuous linear operator on X. We define the quasi-nilpotent part of T as H0(T):={x∈X:lim Note that H_0(T) is a T-invariant linear subspace of X, although perhaps not a closed subspace. Trivially, we have the inclusion \begin{equation*}\mathcal{N}^\infty(T):=\bigcup_{n=1}^\infty\mathcal{N}(T^n)\subseteq H_0(T),\end{equation*} where \mathcal{N}(\cdot) denotes the null … Read more

p-adic analogue of self-adjoint operator

Consider the very well-known result that any Hermitian matrix over C, say T, admits a decomposition T=UDU∗ where U is unitary and D is diagonal with real entries. I am looking for a p-adic analogue: take a matrix A with entries in Qp or some algebraic extension. Is there some condition on A that will … Read more

Spectrum of the hypoelliptic transverse signature operator

Let D be the transverse signature operator constructed by Connes and Moscovici in the paper “Local index formula in Noncommutative Geometry”:this is first order hypoelliptic pseudodifferential operator D defined by the equality D|D|=Q where Q=(dVd∗V−d∗VdV)⊕(dH+d∗H) where dV,dH are vertical and horizontal exterior derivative. It acts on the sections of the bundle Λ(V∗)⊗Λ(p∗(T∗M)) over P:=GL+(M)/SO(n) (the … Read more

A Toeplitz variant of the Hilbert matrix

It is well-known that the Hilbert matrix $H$, i.e., the symmetric Hankel matrix with entries $$H_{m,n}=\frac{1}{m+n-1}, \quad m,n\in\mathbb{N},$$ determines a bounded operator on $\ell^{2}(\mathbb{N})$. Moreover, it is known that $H$ is explicitly diagonalizable which is due to Rosenblum. It seems to be very likely that also the Toeplitz analogue of the Hilbert matrix, i.e., the … Read more

Spectral Gap of Elliptic Operator

Under what conditions on a(x) and domain D, the spectral gap of the elliptic operator ∇⋅(a(x)⋅∇) defined on D, can be controlled? The boundary condition is that the solution at the boundary is zero. Assume that D is a unit ball in Rd. Since the eigenvalues of this operator are countable and nonnegative, the spectral … Read more

Characterizing the separability of the Gelfand space of a semisimple commutative Banach algebra

Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra A equivalent to the existence of a countable family {φn}n∈ω of multiplicative linear functionals on A such that for each a∈A its spectrum coincides with the closure of the union ⋃n∈ωφn(a)? (The problem was posed 09.08.2015 by Michal Wojciechowski on page … Read more

Decay of eigenfunctions for Laplacian

Consider the discrete second derivative with Dirichlet boundary conditions on Cn. Its eigendecomposition is fully known: see wikipedia It seems like the largest eigenvalue λ1 is one with a fast decaying eigenfunction, by this I mean that at the first coordinate |v1,1|≤Cn−3/2. The first 1 indicates the eigenfunction, the second one the coordinate. A priori … Read more

Tight bound on spectral gap of compact homogeneous manifold?

This paper by Peter Li proves a bound on the spectral gap of the Laplacian on a compact homogeneous manifold of diameter d: λ1≥c/d2, where c=π2/4. Can this bound be strengthened in a large number of dimensions, or is it essentially tight? In other words, if we restrict to manifolds of dimension N what is … Read more