Search results for "resolvent"
showing 10 items of 32 documents
Analysis of nonlinear time-dependent properties of carbon fiber reinforced plastic under off-axis loading
2021
Abstract Polymeric composites are rheonomic materials and their deformation can be described using the hereditary elasticity relations which allow for describing the mechanical behavior under time-variable loading with consideration of the influence of temperature and other operational factors. A system of hereditary-type constitutive relations is proposed for off-axis specimens of a unidirectional carbon fiber-reinforced plastic subjected to loading at different strain rates. Using the algebra of resolvent operators and inverted transformation, the constitutive equations allowing of description of anisotropy of rheological properties and, in particular, sensitivity to strain rates are deri…
Dressed emitters as impurities
2021
Dressed states forming when quantum emitters or atoms couple to a photonic bath underpin a number of phenomena and applications, in particular dispersive effective interactions occurring within photonic bandgaps. Here, we present a compact formulation of the resolvent-based theory for calculating atom-photon dressed states built on the idea that the atom behaves as an effective impurity. This establishes an explicit connection with the standard impurity problem in condensed matter. Moreover, it allows us to formulate and settle in a model-independent context a number of properties previously known only for specific models or not entirely formalized. The framework is next extended to the cas…
Perturbations of Jordan Blocks
2019
In this chapter we shall study the spectrum of a random perturbation of the large Jordan block A0, introduced in Sect. 2.4: $$\displaystyle A_0=\begin {pmatrix}0 &1 &0 &0 &\ldots &0\\ 0 &0 &1 &0 &\ldots &0\\ 0 &0 &0 &1 &\ldots &0\\ . &. &. &. &\ldots &.\\ 0 &0 &0 &0 &\ldots &1\\ 0 &0 &0 &0 &\ldots &0 \end {pmatrix}: {\mathbf {C}}^N\to {\mathbf {C}}^N. $$ Zworski noticed that for every z ∈ D(0, 1), there are associated exponentially accurate quasimodes when N →∞. Hence the open unit disc is a region of spectral instability. We have spectral stability (a good resolvent estimate) in \(\mathbf {C}\setminus \overline {D(0,1)}\), since ∥A0∥ = 1. σ(A0) = {0}.
A Note on Riesz Bases of Eigenvectors of Certain Holomorphic Operator-Functions
2001
Abstract Operator-valued functions of the form A (λ) ≔ A − λ + Q(λ) with λ ↦ Q(λ)(A − μ)− 1 compact-valued and holomorphic on certain domains Ω ⊂ C are considered in separable Hilbert space. Assuming that the resolvent of A is compact, its eigenvalues are simple and the corresponding eigenvectors form a Riesz basis for H of finite defect, it is shown that under certain growth conditions on ‖Q(λ)(A − λ)− 1‖ the eigenvectors of A corresponding to a part of its spectrum also form a Riesz basis of finite defect. Applications are given to operator-valued functions of the form A (λ) = A − λ + B(λ − D)− 1C and to spectral problems in L2(0, 1) of the form −f″(x) + p(x, λ)f′(x) + q(x, λ)f(x) = λf(x…
Nonlinear diffusion in transparent media: the resolvent equation
2017
Abstract We consider the partial differential equation u - f = div ( u m ∇ u | ∇ u | ) u-f=\operatornamewithlimits{div}\biggl{(}u^{m}\frac{\nabla u}{|\nabla u|}% \biggr{)} with f nonnegative and bounded and m ∈ ℝ {m\in\mathbb{R}} . We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ℋ N - 1 {{\mathcal{H}}^{N-1}} -Hausdorff measure. Results and proofs extend to more general nonlinearities.
Domains of accretive operators in Banach spaces
2016
LetD(A)be the domain of anm-accretive operatorAon a Banach spaceE. We provide sufficient conditions for the closure ofD(A)to be convex and forD(A)to coincide withEitself. Several related results and pertinent examples are also included.
Semigroups of composition operators and integral operators in spaces of analytic functions
2013
We study the maximal spaces of strong continuity on BMOA and the Bloch space B for semigroups of composition operators. Characterizations are given for the cases when these maximal spaces are V MOA or the little Bloch B0. These characterizations are in terms of the weak compactness of the resolvent function or in terms of a specially chosen symbol g of an integral operator Tg. For the second characterization we prove and use an independent result, namely that the operators Tg are weakly compact on the above mentioned spaces if and only if they are compact.
Operators Which Do Not Have the Single Valued Extension Property
2000
Abstract In this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator T ∈ L ( X ) on a Banach space X and some quantities associated with T which play an important role in Fredholm theory. In particular, we shall consider some conditions for which T does not have the single valued extension property at a point λ o ∈ C .
A-Codes from Rational Functions over Galois Rings
2006
In this paper, we describe authentication codes via (generalized) Gray images of suitable codes over Galois rings. Exponential sums over these rings help determine--or bound--the parameters of such codes.
A non-linear version of Hunt-Lion's theorem from the point of view of T-accretivity
1992
In the classical topological context, Dellacherie [10] has given a non-linear version of Hunt's theorem characterizing the proper kernels verifying the complete maximum principle as those closing a submarkovian resolvent. In this paper we study the relation between this non-linear version of Hunt's theorem and T-accretivity.