Approximating Spectral Invariants of Harper Operators on Graphs

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada (Sun). A main result in this paper is that the spectral density function of DMLs associated to rational weight functions on graphs with a free action of an amenable discrete group can be approximated by the average spectral density function of the DMLs on a regular exhaustion, with either Dirichlet or Neumann boundary conditions. This then gives a criterion for the existence of gaps in the spectrum of the DML, as well as other interesting spectral properties of such DMLs. The technique used incorporates some results of algebraic number theory.     

Pseudodifferential Operators on Periodic Graphs

The main aim of the paper is Fredholm properties of a class of bounded linear operators acting on weighted Lebesgue spaces on an infinite metric graph Γ which is periodic with respect to the action of the group \mathbb Zⁿ{{\mathbb {Z}}^n} . The operators under consideration are distinguished by their local behavior: they act as (Fourier) pseudodifferential operators in the class OPS ⁰ on every open edge of the graph, and they can be represented as a matrix Mellin pseudodifferential operator on a neighborhood of every vertex of Γ. We apply these results to study the Fredholm property of a class of singular integral operators and of certain locally compact operators on graphs.     

Hermitian Adjacency Matrix of Digraphs and Mixed Graphs

The article gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from x to y is equal to the complex unity i (and its symmetric entry is ) if the reverse arc is not present. We also allow arcs in both directions and unoriented edges, in which case we use 1 as the entry. This allows to use the definition also for mixed graphs. This matrix has many nice properties; it has real eigenvalues and the interlacing theorem holds for a digraph and its induced subdigraphs. Besides covering the basic properties, we discuss many differences from the properties of eigenvalues of undirected graphs and develop basic theory. The main novel results include the following. Several surprising facts are discovered about the spectral radius; some consequences of the interlacing property are obtained; operations that preserve the spectrum are discussed—they give rise to a large number of cospectral digraphs; for every , all digraphs whose spectrum is contained in the interval are determined.     

Weighted Norm Inequalities for Spectral Multipliers on Graphs

Let Γ be a graph endowed with a reversible Markov kernel p, whose associated operator P is defined by \(Pf(x) = {\sum }_{y} p(x, y)f(y)\). We assume that the kernels p_n(x, y) associated to Pⁿ satisfy Gaussian upper bounds but do not assume they satisfy the Hölder continuity property and the temporal regularity. Denote by L = I ? P the discrete Laplacian on Γ. This article shows the weighted weak type (1, 1) estimates and the weighted L^p norm inequalities for the spectral multipliers of L. We also obtain the weighted L^p norm inequalities for the commutators of the spectral multipliers of L with BMO functions which are new even for the unweighted case.     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

Essential Spectral Inclusions for Operators on Banach Spaces

This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular, if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators with Bishop’s property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that, under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property plays an essential role in the preservation of the index in this context.      

Differential Operators on Graphs and Photonic Crystals

Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: –u=u, where (x) (the dielectric constant) is a periodic function that assumes a large value near a periodic graph in R ² and is equal to 1 otherwise. High contrast regimes lead to appearence of pseudo-differential operators of the Dirichlet-to-Neumann type on graphs. The paper contains a technique of approximating these pseudo-differential spectral problems by much simpler differential ones that can sometimes be resolved analytically. Numerical experiments show amazing agreement between the spectra of the pseudo-differential and differential problems.     

Spectral Multipliers for Multidimensional Bessel Operators

In this paper we prove L ^p -boundedness properties of spectral multipliers associated with multidimensional Bessel operators. In order to do this we estimate the L ^p -norm of the imaginary powers of Bessel operators. We also prove that the Hankel multipliers of Laplace transform type on (0,∞) ⁿ are principal value integral operators of weak type (1,1).     

On Spectral Hypergraph Theory of the Adjacency Tensor

We study both H and E/Z-eigenvalues of the adjacency tensor of a uniform multi-hypergraph and give conditions for which the largest positive H or Z-eigenvalue corresponds to a strictly positive eigenvector. We also investigate when the E-spectrum of the adjacency tensor is symmetric.     

On the p-Ranks of the Adjacency Matrices of Distance-Regular Graphs

Let be a distance-regular graph with adjacency matrix A. Let I be the identity matrix and J the all-1 matrix. Let p be a prime. We study the p-rank of the matrices A + bJ – cI for integral b, c and the p-rank of corresponding matrices of graphs cospectral with .Using the minimal polynomial of A and the theory of Smith normal forms we first determine which p-ranks of A follow directly from the spectrum and which, in general, do not. For the p-ranks that are not determined by the spectrum (the so-called relevant p-ranks) of A the actual structure of the graph can play a rôle, which means that these p-ranks can be used to distinguish between cospectral graphs.We study the relevant p-ranks for some classes of distance-regular graphs and try to characterize distance-regular graphs by their spectrum and some relevant p-rank.     

Metric Uniformization and Spectral Bounds for Graphs

We present a method for proving upper bounds on the eigenvalues of the graph Laplacian. A main step involves choosing an appropriate “Riemannian” metric to uniformize the geometry of the graph. In many interesting cases, the existence of such a metric is shown by examining the combinatorics of special types of flows. This involves proving new inequalities on the crossing number of graphs.     

Operators on Partial Inner Product Spaces: Towards a Spectral Analysis

Mediterranean Journal of Mathematics - Given a Lattice of Hilbert spaces V J and a symmetric operator A in V J , in the sense of partial inner product spaces, we define a generalized resolvent for...     

Spectral Inclusions for Semigroups of Closed Operators

We show that several spectral inclusions known for C0-semigroups fail for semigroups of closed operators, even if they can be regularized. We introduce the notion of spectral completeness for the regularizing operator C which implies equality of the spectrum and the C-spectrum of the generator. We prove spectral inclusions under this additional assumption. We give a series of examples in which the regularizing operator is spectrally complete including generators of integrated semigroups, of distribution semigroups, and of some semigroups that are strongly continuous for t > 0.     

A Spectral Representation for Bounded Non-Selfadjoint Operators

An integral representation for an arbitrary bounded operator T defined on a Hilbert space ${\mathcal{H}}$ is given. The representing measure is in general defined on a Jordan curve surrounding the spectrum of T. It is obtained as a limit, in a certain weak sense, of a family (F _r ) of absolutely continuous measures the Radon?CNikodym derivative of which (with respect to the standard Lebesgue measure on the considered Jordan curve) are described explicitly in terms of the operator T and its adjoint T ^*.     

Isoenergy Spectral Problem for Multidimensional Difference Operators

In this paper, the direct and inverse isoenergy spectral problems are solved for a class of multidimensional periodic difference operators. It is proved that the inverse spectral problem is solvable in terms of theta functions of curves added to the spectral variety under compactification, and multidimensional analogs of the Veselov–Novikov relations are found.