Notion Of Spectrum Research Articles

Nonuniform μ-dichotomy spectrum and kinematic similarity

For linear nonautonomous differential equations we introduce a new family of spectrums defined with general nonuniform dichotomies: for a given growth rate μ in a large family of growth rates, we consider a notion of spectrum, named nonuniform μ-dichotomy spectrum. This family of spectrums contain the nonuniform dichotomy spectrum as the very particular case of exponential growth rates. For each growth rate μ, we describe all possible forms of the nonuniform μ-dichotomy spectrum, relate its connected components with adapted notions of Lyapunov exponents, and use it to obtain a reducibility result for nonautonomous linear differential equations. We also give illustrative examples where the spectrum is obtained, including a situation where a normal form is obtained for polynomial behavior.

Higher order spectrum and quasi-linear boundary problems of discrete p-Laplacian

In the study of problems of the type , conditions are generally imposed on the asymptotic behaviour of with respect to the spectrum of . To study the quasi-linear problems type, we introduce a notion of spectrum for that takes into account the dependence of the right-hand side with respect to the gradient. The object of this work is to define, study and apply this new notion of spectrum.

Schatten class and Berezin transform of quaternionic linear operators

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S‐spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S‐functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.

The H∞ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators

In this paper we extend the H∞ functional calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S-functional calculus. The S-functional calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford functional calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-functional calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H∞ functional calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H∞ functional calculus for (n+1)-tuples of operators applies, in particular, to the Dirac operator.

The Spectral Theorem for Unitary Operators Based on the S-Spectrum

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutative versions of the Riesz-Dunford functional calculus. The main tools to prove the spectral theorem for unitary operators are the quaternionic version of Herglotz's theorem, which relies on the new notion of $q$-positive measure, and quaternionic spectral measures, which are related to the quaternionic Riesz projectors defined by means of the $S$-resolvent operator and the $S$-spectrum. The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the spectral theorem.

Real Powers of Bounded Linear Operators

This paper introduces \((I-A)^{\alpha }\) for every real number \(\alpha ,\) when A is a bounded linear operator on a Banach space \({\mathbb {X}}\) with norm of A less than 1. The principal value of \((I-A)^{\alpha }\) is defined as the limit of a series in the operator norm topology. Convergence of the series is proved using the Newton’s binomial theorem and this can be regarded as a noncommutative analogue of binomial theorem. Use this to introduce the generalized notion of spectrum of a bounded operator and finally, propose a method for finding some square roots of matrices in \(M_{n}({\mathbb {C}}),\) the set of all \(n\times n\) matrices with complex entries.

The spectra of irreducible matrices over completed idempotent semifields

Motivated by some spectral results in the characterization of concept lattices we investigate the spectra of reducible matrices over complete idempotent semifields in the framework of naturally-ordered semirings, or dioids. We find non-null eigenvectors for every non-null element in the semifield and conclude that the notion of spectrum has to be refined to encompass that of the incomplete semifield case so as to include only those eigenvalues with eigenvectors that have finite coordinates. Considering special sets of eigenvectors brings out finite complete lattices in the picture and we contend that such structure may be more important than standard eigenspaces for matrices over completed idempotent semifields.

The multiplicative spectrum and the uniqueness of the complete norm topology

We define the spectrum of an element a in a non-associative algebra A according to a classical notion of invertibility (a is invertible if the multiplication operators La and Ra are bijective). Around this notion of spectrum, we develop a basic theoretical support for a non-associative spectral theory. Thus we prove some classical theorems of automatic continuity free of the requirement of associativity. In particular, we show the uniqueness of the complete norm topology of m-semisimple algebras, obtaining as a corollary of this result a well-known theorem of Barry E. Johnson (1967). The celebrated result of C.E. Rickart (1960) about the continuity of dense-range homomorphisms is also studied in the non-associative framework. Finally, because non-associative algebras are very suitable models in genetics, we provide here a hint of how to apply this approach in that context, by showing that every homomorphism from a complete normed algebra onto a particular type of evolution algebra is automatically continuous.

The Fueter mapping theorem in integral form and the ℱ-functional calculus

In this paper we show a version of the Fueter mapping theorem that can be stated in integral form based on the Cauchy formulas for slice monogenic (or slice regular) functions. More precisely, given a holomorphic function f of a paravector variable, we generate a monogenic function by an integral transform whose kernel is particularly simple. This procedure allows us to define a functional calculus for n-tuples of commuting operators (called ℱ-functional calculus) based on a new notion of spectrum, called ℱ-spectrum, for the n-tuples of operators. Analogous results are shown for the quaternionic version of the theory and for the related ℱ-functional calculus. Copyright © 2010 John Wiley & Sons, Ltd.

Spectral theory for non-associative complete normed algebras and automatic continuity

This paper concerns normed algebras whose product is free of the requirement of associativity. For these algebras, a very natural notion of spectrum is provided and basic spectral properties of (associative) Banach algebras are extended to this general non-associative setting. These results are applied to generalize Rickart's dense-range homomorphism theorem. Our approach sheds some light on the classical associative theory.

About Stone's notion of spectrum

This paper analyses two fundamental notes by Stone, giving two ways of representing a compact space in term of some algebra of functions over this space, in the framework of point-free topology. As applications, we give an alternative approach to the spectral theorem, and we present constructive proofs of results of Krivine and of the Kadison-Dubois theorem.

Orders, reduction graphs and spectra

In this paper, we consider Abstract Reduction Systems as the setting where we need to investigate order properties of reduction graphs. To this aim, the main tool is the notion of spectrum, which captures some essential features of reduction sequences. We show that each spectrum is a complete partial order with respect to a suitable information ordering. Then we consider linearly ordered spectra. For those of them which are not well-ordered, we obtain a graph-theoretic characterization in terms of forbidden subgraphs. For well-ordered spectra, we show that they can represent all countable successor ordinals. Then, considering constructive ordinals, we address the well-known problem of knowing which ordinals are lambda representable (that is for which ordinal α there exists a lambda term T such that α is isomorphic to the spectrum of T). We give a partial answer by showing that all successor ordinals a, with α < ε 0, are lambda-representable.

Gabriel topologies on coherent quantales

The set of Gabriel topologies on a coherent quantale ordered under inclusion is a frame (studied by Rosenthal, Simmons and others). The set of all those Gabriel topologies that are inaccessible by directed joins (we call such topologies compact) is a subframe of it. When the quantale under consideration is commutative the frame of compact topologies is coherent. Several notions of spectra in ring theory appear as instances of this construction. When the quantale is non-commutative and coherent and its finite elements are closed under (right) implication then the frame of compact topologies is locally compact and compact. We present an interpretation of the notion of compact Gabriel topology on a coherent quantale in terms of deductively closed sets of formulae for a system of prepositional logic without the contraction and possibly the exchange rule (but admitting weakening). Our local compactness (and the subsequent spatiality) results for the frame of compact topologies correspond to a completeness theorem for such a system.

Conditional resonance-fluorescence spectra of single atoms

The temporal behavior of a single quantum system may contain information, for instance, about macroscopic dark periods in electron shelving, which is washed out or even lost in ensemble averages. The same is true in principle for spectral information. For example, for an atom with light and dark periods one might try to spectrally analyze the light emitted in a light period of given length or, more generally, the light emitted into one solid angle while monitoring the temporal emission in another solid angle. Considering different emission behaviors one may get different spectra, i.e., spectra that are dependent on the conditions imposed. In this paper, such questions, in particular the notion of spectrum in a light period, are studied. To this end we generalize the recent quantum jump approach to include resonance-fluorescence spectra of single atoms. We explicitly show that the theory developed in this paper easily generalizes, e.g., to absorption spectra. We apply the theory to a V system that has one metastable level and that may exhibit intermittent fluorescence when driven by two lasers. We show that the spectrum of resonance fluorescence of the complete ensemble differs substantially from that observed in a light period. A physical explanation of this behavior is presented.

The Spectra of Infinite Hypertrees

A model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs. Two notions of spectra, or “first eigenvalue,” are then examined for the infinite tree, obtaining a precise value for the first notion and obtaining some estimates for the second. The results indicate agreement of the first eigenvalue of the infinite hypertree with the ”second eigenvalue“ of a random hypergraph of the same degree, to within logarithmic factors, at least for the first notion of first eigenvalue.

Generalized Spectral Theory in Complex Banach Algebras

Let A be an element of a complex Banach algebra with identitI. The ordinary spectrum of A, sp(A), consists of those points z in the complex plane such that A — zI has no inverse in . If Q is any other element of , we define spQ(A), the spectrum of A relative to Q, or Q-spectrum of A, as those points z such that has no inverse in . Thus if Q = 0 the Q-spectrum of A is the same as the ordinary spectrum of A.The generalized notion of spectrum, spQ(A), retains many of the properties of the ordinary spectrum, particularly when A and Q commute and the ordinary spectrum of Q does not meet the unit circle. Under these conditions the Q-spectrum of A is a nonempty compact subset of the plane, and if both sp(A) and sp(Q) are finite (or countable), so is spQ(A).

MODULI OF STABLE VECTOR BUNDLES OF RANK 2 ON $ P_3$ WITH FIXED SPECTRUM

Moduli of stable rank 2 vector bundles on the three-dimensional complex projective space are studied. Using the notion of spectrum, two series of components in the moduli scheme of such bundles are pointed out. Each component is an irreducible rational variety, and asymptotically the dimension of the components behaves as a quadratic or cubic polynomial. The monadic realization of a generic bundle on the components is given by a rank one monad. Bibliography: 5 titles.

Recovering a frame from its sheaves of algebras

Let ▪ be a frame, α an element of ▪ and T a finitary algebraic theory. In this paper we compare the category ▪ of sheaves of T-algebras on ▪ with the category Sh(α↓, T) of sheaves of T-algebras on α↓ (where α↓ is the initial segment {β|β<α}). This comparison suggests the definition of formal initial segments of the category ▪. For a large class of theories to be called ‘integral’ (examples of which are sets, monoids, groups, rings, modules on a integral domain, boolean algebras,hellip;) the formal initial segments of ▪ coincide with the usual initial segments of ▪: this means that we are able to recover ▪ axiomatically from ▪. When ▪ is the initial frame {0, 1}, the frame of formal initial segments of ▪ is the frame of open subsets of a compact space Spec T, called the spectrum of the theory T. When T is the theory of modules on some ring R, we recover in this way various well known notions of spectra and their corresponding sheaf-representation of the ring.

�ber nichtlineare Spektral- und St�rungstheorie

A topology is defined on the set of closed operators between two Banach spaces. We show that the set of Lipschitzcontinuous operators is open in this topology; the relative topology on this subset is the natural one. The notion of spectrum is defined for nonlinear maps, and we prove among others the following facts: the resolvent set is open and depends upper semi-continuously on the operator; for Lipschitzcontinuous operators, the spectrum is compact; the resolvent map is continuous. Then we examine more closely the cases of Frechet-differentiable maps and of isolated points in the spectrum.