Generalized spectral perturbation and the boundary spectrum

Sonja Mouton

Displaying similar documents to “Generalized spectral perturbation and the boundary spectrum”

Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk (2018)

Mathematica Bohemica

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We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $𝒜$ to have a non-void left joint spectrum $σ_{l} (S) .$ In particular, for a Lie subalgebra $L \subset 𝒜$ we have $σ_{l} (L) \neq \emptyset$ if and only if $[L, L]$ generates in $𝒜$ a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

The third order spectrum of the p-biharmonic operator with weight

Khalil Ben Haddouch, Najib Tsouli, Zakaria El Allali (2014)

Applicationes Mathematicae

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We show that the spectrum of $Δ ²_{p} u + {2 β \cdot \nabla (| Δ u |}^{p - 2} {Δ u) + | β | ² | Δ u |}^{p - 2} Δ u = α m {| u |}^{p - 2} u$ , where $β \in ℝ^{N}$ , under Navier boundary conditions, contains at least one sequence of eigensurfaces.

Some examples of cocycles with simple continuous singular spectrum

K. Frączek (2001)

Studia Mathematica

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We study spectral properties of Anzai skew products $T_{φ} : ² \to ²$ defined by $T_{φ} (z, ω) = (e^{2 π i α} z, φ (z) ω)$ , where α is irrational and φ: → is a measurable cocycle. Precisely, we deal with the case where φ is piecewise absolutely continuous such that the sum of all jumps of φ equals zero. It is shown that the simple continuous singular spectrum of $T_{φ}$ on the orthocomplement of the space of functions depending only on the first variable is a “typical” property in the above-mentioned class of cocycles, if α admits a sufficiently...

Resonant delocalization for random Schrödinger operators on tree graphs

Michael Aizenman, Simone Warzel (2013)

Journal of the European Mathematical Society

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We analyse the spectral phase diagram of Schrödinger operators $T + λ V$ on regular tree graphs, with $T$ the graph adjacency operator and $V$ a random potential given by $i i d$ random variables. The main result is a criterion for the emergence of absolutely continuous $(a c)$ spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials $a c$ spectrum appears at arbitrarily weak disorder $(λ ≪ 1)$ in an energy regime which extends beyond the spectrum of $\sim T$ ....

On the convergence and character spectra of compact spaces

István Juhász, William A. R. Weiss (2010)

Fundamenta Mathematicae

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An infinite set A in a space X converges to a point p (denoted by A → p) if for every neighbourhood U of p we have |A∖U| < |A|. We call cS(p,X) = |A|: A ⊂ X and A → p the convergence spectrum of p in X and cS(X) = ⋃cS(x,X): x ∈ X the convergence spectrum of X. The character spectrum of a point p ∈ X is χS(p,X) = χ(p,Y): p is non-isolated in Y ⊂ X, and χS(X) = ⋃χS(x,X): x ∈ X is the character spectrum of X. If κ ∈ χS(p,X) for a compactum X then κ,cf(κ) ⊂ cS(p,X). A selection of our...

Graphs with small diameter determined by their $D$ -spectra

Ruifang Liu, Jie Xue (2018)

Czechoslovak Mathematical Journal

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Let $G$ be a connected graph with vertex set $V (G) = {v_{1}, v_{2}, ..., v_{n}}$ . The distance matrix $D (G) = {(d_{i j})}_{n \times n}$ is the matrix indexed by the vertices of $G$ , where $d_{i j}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$ . Suppose that $λ_{1} (D) \geq λ_{2} (D) \geq \dots \geq λ_{n} (D)$ are the distance spectrum of $G$ . The graph $G$ is said to be determined by its $D$ -spectrum if with respect to the distance matrix $D (G)$ , any graph having the same spectrum as $G$ is isomorphic to $G$ . We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs...

On generalized property (v) for bounded linear operators

J. Sanabria, C. Carpintero, E. Rosas, O. García (2012)

Studia Mathematica

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An operator T acting on a Banach space X has property (gw) if $σ_{a} (T) ∖ σ_{S B F ₊ ¯} (T) = E (T)$ , where $σ_{a} (T)$ is the approximate point spectrum of T, $σ_{S B F ₊ ¯} (T)$ is the upper semi-B-Weyl spectrum of T and E(T) is the set of all isolated eigenvalues of T. We introduce and study two new spectral properties (v) and (gv) in connection with Weyl type theorems. Among other results, we show that T satisfies (gv) if and only if T satisfies (gw) and $σ (T) = σ_{a} (T)$ .

Borel parts of the spectrum of an operator and of the operator algebra of a separable Hilbert space

Piotr Niemiec (2012)

Studia Mathematica

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For a linear operator T in a Banach space let $σ_{p} (T)$ denote the point spectrum of T, let $σ_{p, n} (T)$ for finite n > 0 be the set of all $λ \in σ_{p} (T)$ such that dim ker(T - λ) = n and let $σ_{p, \infty} (T)$ be the set of all $λ \in σ_{p} (T)$ for which ker(T - λ) is infinite-dimensional. It is shown that $σ_{p} (T)$ is $ℱ_{σ}$ , $σ_{p, \infty} (T)$ is $ℱ_{σ δ}$ and for each finite n the set $σ_{p, n} (T)$ is the intersection of an $ℱ_{σ}$ set and a $_{δ}$ set provided T is closable and the domain of T is separable and weakly σ-compact. For closed densely defined operators in a separable Hilbert space a more...

Weighted Frobenius-Perron operators and their spectra

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2017)

Mathematica Bohemica

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First, some classic properties of a weighted Frobenius-Perron operator $𝒫_{ϕ}^{u}$ on $L^{1} (Σ)$ as a predual of weighted Koopman operator $W = u U_{ϕ}$ on $L^{\infty} (Σ)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of $𝒫_{ϕ}^{u}$ under certain conditions.

Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee, Quoc-Phong Vu (2014)

Studia Mathematica

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We consider simultaneous solutions of operator Sylvester equations $A_{i} X - X B_{i} = C_{i}$ (1 ≤ i ≤ k), where $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C ₁, . . ., C_{k})$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

The fan graph is determined by its signless Laplacian spectrum

Muhuo Liu, Yuan Yuan, Kinkar Chandra Das (2020)

Czechoslovak Mathematical Journal

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Given a graph $G$ , if there is no nonisomorphic graph $H$ such that $G$ and $H$ have the same signless Laplacian spectra, then we say that $G$ is $Q$ -DS. In this paper we show that every fan graph $F_{n}$ is $Q$ -DS, where $F_{n} = K_{1} \lor P_{n - 1}$ and $n \geq 3$ .

The essential spectrum of holomorphic Toeplitz operators on $H^{p}$ spaces

Mats Andersson, Sebastian Sandberg (2003)

Studia Mathematica

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We compute the essential Taylor spectrum of a tuple of analytic Toeplitz operators $T_{g}$ on $H^{p} (D)$ , where D is a strictly pseudoconvex domain. We also provide specific formulas for the index of $T_{g}$ provided that $g^{- 1} (0)$ is a compact subset of D.

Fredholm spectrum and growth of cohomology groups

Jörg Eschmeier (2008)

Studia Mathematica

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Let T ∈ L(E)ⁿ be a commuting tuple of bounded linear operators on a complex Banach space E and let $σ_{F} (T) = σ (T) ∖ σ_{e} (T)$ be the non-essential spectrum of T. We show that, for each connected component M of the manifold $R e g (σ_{F} (T))$ of all smooth points of $σ_{F} (T)$ , there is a number p ∈ 0, ..., n such that, for each point z ∈ M, the dimensions of the cohomology groups $H^{p} ({(z - T)}^{k}, E)$ grow at least like the sequence ${(k^{d})}_{k \geq 1}$ with d = dim M.

Schur Lemma and the Spectral Mapping Formula

Antoni Wawrzyńczyk (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula $σ_{l} (S)$ = ${(λ (s))}_{s \in S} \in ℂ^{S} | {s - λ (s)}_{s \in S}$ generates a proper left ideal $.$ Using the Schur lemma and the Gelfand-Mazur theorem we prove that $σ_{l} (S)$ has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.

On the spectrum of the operator which is a composition of integration and substitution

Ignat Domanov (2008)

Studia Mathematica

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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{ϕ} : f (x) \mapsto \int_{0}^{ϕ (x)} f (t) d t$ be defined on L₂[0,1]. We prove that $V_{ϕ}$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{ϕ}$ always equals 1.

On the perturbation functions and similarity orbits

Haïkel Skhiri (2008)

Studia Mathematica

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We show that the essential spectral radius $ϱ_{e} (T)$ of T ∈ B(H) can be calculated by the formula $ϱ_{e} (T)$ = inf $ℱ_{♯ \cdot ♯} (X T X^{- 1})$ : X an invertible operator, where $ℱ_{♯ \cdot ♯} (T)$ is a Φ₁-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if $_{♯ \cdot ♯} (T)$ is a Φ₂-perturbation function [loc. cit.] and if T is a Fredholm operator, then $d i s t (0, σ_{e} (T))$ = sup $_{♯ \cdot ♯} (X T X^{- 1})$ : X an invertible operator.

Proximality in Pisot tiling spaces

Marcy Barge, Beverly Diamond (2007)

Fundamenta Mathematicae

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A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space $_{φ}$ has pure discrete spectrum, we describe the collection $_{φ}^{P}$ of pairs of proximal tilings in $_{φ}$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then $_{φ}$ and $_{ψ}$ are homeomorphic...

Properties of Wiener-Wintner dynamical systems

I. Assani, K. Nicolaou (2001)

Bulletin de la Société Mathématique de France

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In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if $f \in L^{p}$ , $p$ large enough, is a Wiener-Wintner function then, for all $γ \in (1 + \frac{1}{2 p} - \frac{β}{2}, 1]$ , there exists a set $X_{f}$ of full measure for which the series $\sum_{n = 1}^{\infty} \frac{f (T^{n} x) e^{2 π i n ϵ}}{n^{γ}}$ converges uniformly with respect to $ϵ$ .

A-Browder-type theorems for direct sums of operators

Mohammed Berkani, Mustapha Sarih, Hassan Zariouh (2016)

Mathematica Bohemica

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We study the stability of a-Browder-type theorems for orthogonal direct sums of operators. We give counterexamples which show that in general the properties $(SBaw)$ , $(SBab)$ , $(SBw)$ and $(SBb)$ are not preserved under direct sums of operators. However, we prove that if $S$ and $T$ are bounded linear operators acting on Banach spaces and having the property $(SBab)$ , then $S \oplus T$ has the property $(SBab)$ if and only if $σ_{{SBF}_{+}^{-}} (S \oplus T) = σ_{{SBF}_{+}^{-}} (S) \cup σ_{{SBF}_{+}^{-}} (T)$ , where $σ_{{SBF}_{+}^{-}} (T)$ is the upper semi-B-Weyl spectrum of $T$ . We obtain analogous preservation results for the properties $(SBaw)$ ,...

The spectral determinations of the connected multicone graphs $K_{w} ▽ m P_{17}$ and $K_{w} ▽ m S$

Ali Zeydi Abdian, S. Morteza Mirafzal (2018)

Czechoslovak Mathematical Journal

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Finding and discovering any class of graphs which are determined by their spectra is always an important and interesting problem in the spectral graph theory. The main aim of this study is to characterize two classes of multicone graphs which are determined by both their adjacency and Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let $K_{w}$ denote a complete graph on $w$ vertices, and let $m$ be a positive integer number. In A. Z. Abdian (2016)...

Symmetries of the nonlinear Schrödinger equation

Benoît Grébert, Thomas Kappeler (2002)

Bulletin de la Société Mathématique de France

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Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum $\dots < λ_{k}^{-} \leq λ_{k}^{+} < λ_{k + 1}^{-} \leq \dots$ of a Zakharov-Shabat operator is symmetric,. $λ_{k}^{\pm} = - λ_{- k}^{\mp}$ for all $k$ , if and only if the sequence ${(γ_{k})}_{k \in ℤ}$ of gap lengths, $γ_{k} : = λ_{k}^{+} - λ_{k}^{-}$ , is symmetric with respect to $k = 0$ .

A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

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We define partial spectral integrals $S_{R}$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $l o g (2 + L_{α}) f \in L ²$ , where $L_{α}$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R} f (x)$ converges a.e. to f(x) as R → ∞.

Horocyclic products of trees

Laurent Bartholdi, Markus Neuhauser, Wolfgang Woess (2008)

Journal of the European Mathematical Society

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Let $T_{1}, \dots, T_{d}$ be homogeneous trees with degrees $q_{1} + 1, \dots, q_{d} + 1 \geq 3$ , respectively. For each tree, let $𝔥 : T_{j} \to ℤ$ be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of $T_{1}, \dots, T_{d}$ is the graph $𝖣𝖫 (q_{1}, \dots, q_{d})$ consisting of all $d$ -tuples $x_{1} \dots x_{d} \in T_{1} \times \dots \times T_{d}$ with $𝔥 (x_{1}) + \dots + 𝔥 (x_{d}) = 0$ , 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...

The density of states of a local almost periodic operator in $ℝ^{ν}$

Andrzej Krupa (2003)

Studia Mathematica

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We prove the existence of the density of states of a local, self-adjoint operator determined by a coercive, almost periodic quadratic form on $H^{m} (ℝ^{ν})$ . The support of the density coincides with the spectrum of the operator in $L ² (ℝ^{ν})$ .

Spectral properties of quotients of Beurling-type submodules of the Hardy module over the unit ball

Kunyu Guo, Yongjiang Duan (2006)

Studia Mathematica

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Let M be a Beurling-type submodule of $H ² (_{d})$ , the Hardy space over the unit ball $_{d}$ of $ℂ^{d}$ , and let $N = H ² (_{d}) / M$ be the associated quotient module. We completely describe the spectrum and essential spectrum of N, and related index theory.

When spectra of lattices of $z$ -ideals are Stone-Čech compactifications

Themba Dube (2017)

Mathematica Bohemica

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Let $X$ be a completely regular Hausdorff space and, as usual, let $C (X)$ denote the ring of real-valued continuous functions on $X$ . The lattice of $z$ -ideals of $C (X)$ has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) $β X$ precisely when $X$ is a $P$ -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a $d$ -ideal if whenever two elements have the same annihilator...

Coincidence for substitutions of Pisot type

Marcy Barge, Beverly Diamond (2002)

Bulletin de la Société Mathématique de France

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Let $ϕ$ be a substitution of Pisot type on the alphabet $𝒜 = {1, 2, ..., d}$ ; $ϕ$ satisfies theif for every $i, j \in 𝒜$ , there are integers $k, n$ such that $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same $k$ -th letter, and the prefixes of length $k - 1$ of $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d = 2$ and provide a partial result for $d \geq 2$ .