This paper is concerned with an extension and reinterpretation of previous results on the variational characterization of eigenvalues in gaps of the essential spectrum of self-adjoint operators. We state two general abstract results on the existence of eigenvalues in the gap and a continuation principle. Then these results are applied to Dirac operators in order to characterize simultaneously eigenvalues corresponding to electronic and positronic bound states.

An improvement of the generalization-obtained in a previous article [Bu1] by the author-of the uniform ergodic theorem to poles of arbitrary order is derived. In order to answer two natural questions suggested by this result, two examples are also given. Namely, two bounded linear operators T and A are constructed such that $n^{-2}Tⁿ$ converges uniformly to zero, the sum of the range and the kernel of 1-T being closed, and $n^{-3}{\sum }_{k=0}^{n-1}{A}^k$ converges uniformly, the sum of the range of 1-A and the kernel of (1-A)² being...

Let T be a bounded linear operator acting on a Banach space X such that T or T* has the single-valued extension property (SVEP). We prove that the spectral mapping theorem holds for the semi-essential approximate point spectrum σSBF-+(T); and we show that generalized a-Browder's theorem holds for f(T) for every analytic function f defined on an open neighbourhood U of σ(T): Moreover, we give a necessary and sufficient condition for such T to obey generalized a-Weyl's theorem. An application is given...

By considering arbitrary mappings $\omega$ from a Banach algebra $A$ into the set of all nonempty, compact subsets of the complex plane such that for all $a\in A$, the set $\omega \left(a\right)$ lies between the boundary and connected hull of the exponential spectrum of $a$, we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover a number of new properties of the boundary spectrum.

A bounded operator T ∈ L(X) acting on a Banach space X is said to satisfy generalized Weyl's theorem if the complement in the spectrum of the B-Weyl spectrum is the set of all eigenvalues which are isolated points of the spectrum. We prove that generalized Weyl's theorem holds for several classes of operators, extending previous results of Istrăţescu and Curto-Han. We also consider the preservation of generalized Weyl's theorem between two operators T ∈ L(X), S ∈ L(Y) intertwined or asymptotically...

According to the von Neumann-Halperin and Lapidus theorems, in a Hilbert space the iterates of products or, respectively, of convex combinations of orthoprojections are strongly convergent. We extend these results to the iterates of convex combinations of products of some projections in a complex Banach space. The latter is assumed uniformly convex or uniformly smooth for the orthoprojections, or reflexive for more special projections, in particular, for the hermitian ones. In all cases the proof...

Let $A$ be a unital Banach algebra over $ℂ$, and suppose that the nonzero spectral values of $a$ and $b\in A$ are discrete sets which cluster at $0\in ℂ$, if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that $a$ and $b$ are quasinilpotent equivalent if...

Let (X) be the algebra of all bounded operators on a Banach space X, and let θ: G → (X) be a strongly continuous representation of a locally compact and second countable abelian group G on X. Set σ¹(θ(g)): = λ/|λ| | λ ∈ σ(θ(g)), where σ(θ(g)) is the spectrum of θ(g), and let ${\Sigma }_{\theta }$ be the set of all g ∈ G such that σ¹(θ(g)) does not contain any regular polygon of (by a regular polygon we mean the image under a rotation of a closed subgroup of the unit circle different from 1). We prove that θ is uniformly...

A Banach space operator T belonging to B(X) is said to be hereditarily normaloid, T ∈ HN, if every part of T is normaloid; T ∈ HN is totally hereditarily normaloid, T ∈ THN, if every invertible part of T is also normaloid; and T ∈ CHN if either T ∈ THN or T - λI is in HN for every complex number λ. Class CHN is large; it contains a number of the commonly considered classes of operators. We study operators T ∈ CHN, and prove that the Riesz projection associated with a λ ∈ isoσ(T), T ∈ CHN ∩ B(H)...

Let $T_1,\cdots ,T_d$ be homogeneous trees with degrees $q_1+1,\cdots ,q_d+1\ge 3$, respectively. For each tree, let $𝔥:T_j\to \mathbb{Z}$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_1,\cdots ,T_d$ is the graph $\mathrm{𝖣𝖫}(q_1,\cdots ,q_d)$ consisting of all $d$-tuples $x_1\cdots x_d\in T_1\times \cdots \times T_d$ with $𝔥\left(x_1\right)+\cdots +𝔥\left(x_d\right)=0$, equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If $d=2$ and $q_1=q_2=q$ then we obtain a Cayley graph of the...

On étudie un opérateur de la forme $-\Delta +V$ sur ${ℝ}^d$, où $V$ est un potentiel admettant plusieurs pôles en $a/r^2$. Plus précisément, on démontre l’estimation de résolvante tronquée à hautes fréquences, classique dans les cas non-captifs, et qui implique l’effet régularisant standard pour l’équation de Schrödinger correspondante. La preuve est basée sur l’introduction d’une mesure de défaut micro-locale semi-classique. On démontre également, dans le même contexte, des inégalités de Strichartz pour l’équation de Schrödinger....