Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk

Displaying similar documents to “Subsets of nonempty joint spectrum in topological algebras”

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...

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.

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.

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...

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 $\sim T$ ....

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 $n \geq 3$ .

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.

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.

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.

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 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 $σ (T) = σ_{a} (T)$ .

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 $(SBaw)$ ,...

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 $ϵ$ .

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...

Enveloping algebras of Slodowy slices and the Joseph ideal

Alexander Premet (2007)

Journal of the European Mathematical Society

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Let $G$ be a simple algebraic group over an algebraically closed field $𝕜$ of characteristic 0, and $𝔤 = Lie G$ . Let $(e, h, f)$ be an $𝔰 𝔩_{2}$ -triple in $𝔤$ with $e$ being a long root vector in $𝔤$ . Let $(\cdot, \cdot)$ be the $G$ -invariant bilinear form on $𝔤$ with $(e, f) = 1$ and let $χ \in 𝔤^{*}$ be such that $χ (x) = (e, x)$ for all $x \in 𝔤$ . Let $𝒮$ be the Slodowy slice at $e$ through the adjoint orbit of $e$ and let $H$ be the enveloping algebra of $𝒮$ ; see [31]. In this article we give an explicit presentation of $H$ by generators and relations. As a consequence we deduce that $H$ contains...

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...

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 $d \geq 2$ .

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...

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...

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...