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This article presents a new proof of a theorem concerning bounds of the spectrum of the product of unitary operators and a generalization for differentiable curves of this theorem. The proofs involve metric geometric arguments in the group of unitary operators and the sphere where these operators act. [ABSTRACT FROM AUTHOR]

We derive several upper bounds on the spectral gap of the Laplacian on compact metric graphs with standard or Dirichlet vertex conditions. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling. [ABSTRACT FROM AUTHOR]

Let G = ∗λ Gλbe a free product of torsion-free groups, and let g ∈ [G,G] be any element not conjugate into a Gλ. Then sclG(g) ≥ 1/2. This generalizes and gives a new proof of a theorem of Duncan and Howie (1991). [ABSTRACT FROM AUTHOR]

We define the complete numerical radius norm for homomorphisms from any operator algebra into B(H), and show that this norm can be computed explicitly in terms of the completely bounded norm. This is used to show that if K is a complete C-spectral set for an operator T, then it is a complete M-numerical radius set, where M = 1/2 (C + C-1). In particular, in view of Crouzeix's theorem, there is a universal constant M (less than 5.6) so that if P is a matrix polynomial and T ∈ B(H), then w(P(T)) ≤ || P || W(T). When W(T) = D, we have M = 5/4. [ABSTRACT FROM AUTHOR]

The purpose of the present work is to answer an open problem which was raised by G. Costakis and A. Manoussos in their paper "J-class operators and hypercyclicity", J. Operator Theory 67 (2012), 101-119. More precisely, we give the spectral description of the closure of the set of J-class operators acting on a separable Hilbert space. [ABSTRACT FROM AUTHOR]

In this paper we extend the results of the research started by the first author in which Karlin-McGregor diagonalization of certain reversible Markov chains over countably infinite general state spaces by orthogonal polynomials was used to estimate the rate of convergence to a stationary distribution. We use a method of Koornwinder to generate a large and interesting family of random walks which exhibits a lack of spectral gap, and a polynomial rate of convergence to the stationary distribution. For the Chebyshev type subfamily of Markov chains, we use asymptotic techniques to obtain an upper bound of order O ( log t/ √t and a lower bound of order O (1/√ t on the distance to the stationary distribution regardless of the initial state. Due to the lack of a spectral gap, these results lie outside the scope of geometric ergodicity theory. [ABSTRACT FROM AUTHOR]

We study the Laplace spectra of the intrinsic instantaneous metrics on the event and cosmological horizons of a Kerr-Newman-de Sitter space-time and prove that the spectral data from these horizons uniquely determine the metric among all such space-times. This is accomplished by exhibiting formulae relating the parameters of the space-time metric to the traces of invariant and equivariant Green's operators associated with these Laplacians. In particular, an interesting explicit formula for the cosmological constant is found. [ABSTRACT FROM AUTHOR]

Let R and S be commuting n-tuples. We give some spectral and local spectral relations between RS and SR. In particular, we show that RS has the single valued extension property or satisfies Bishop's property (β) if and only if SR has the corresponding property. [ABSTRACT FROM AUTHOR]

We study the Laplace operator with Dirichlet or Neumann boundary conditions on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has a simple spectrum. We also address the more general case of geodesic polygons in a constant curvature space form. [ABSTRACT FROM AUTHOR]