On the spectrum and eigenfunctions of the operator $(Vf)(x) = ∫_0^{x^{α}} f(t)dt$

I. Yu. Domanov

Displaying similar documents to “On the spectrum and eigenfunctions of the operator $(V f) (x) = \int_{0}^{x^{α}} f (t) d t$ ”

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.

Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

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We investigate the behaviour of a sequence $λ_{s}$ , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains $Ω_{s}$ , s = 1,2,..., obtained by removing from a given domain Ω a set $E_{s}$ whose diameter vanishes when s → ∞. We estimate the deviation of $λ_{s}$ from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

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.

Monotonicity of first eigenvalues along the Yamabe flow

Liangdi Zhang (2021)

Czechoslovak Mathematical Journal

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We construct some nondecreasing quantities associated to the first eigenvalue of $- Δ_{φ} + c R$ $(c \geq \frac{1}{2} (n - 2) / (n - 1))$ along the Yamabe flow, where $Δ_{φ}$ is the Witten-Laplacian operator with a $C^{2}$ function $φ$ . We also prove a monotonic result on the first eigenvalue of $- Δ_{φ} + \frac{1}{4} (n / (n - 1)) R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $- Δ_{φ} + c R^{a}$ with $a \in (0, 1)$ along the Yamabe flow.

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...

Global continuum of positive solutions for discrete $p$ -Laplacian eigenvalue problems

Dingyong Bai, Yuming Chen (2015)

Applications of Mathematics

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We discuss the discrete $p$ -Laplacian eigenvalue problem, $\{\begin{matrix} Δ (φ_{p} (Δ u (k - 1))) + λ a (k) g (u (k)) = 0, k \in {1, 2, ..., T}, \\ u (0) = u (T + 1) = 0, \end{matrix}$ where $T > 1$ is a given positive integer and $φ_{p} (x) : = {| x |}^{p - 2} x$ , $p > 1$ . First, the existence of an unbounded continuum $𝒞$ of positive solutions emanating from $(λ, u) = (0, 0)$ is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any $λ > 0$ and all solutions are ordered. Thus the continuum $𝒞$ is a monotone continuous curve globally defined for all $λ > 0$ .

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...

Estimates of the principal eigenvalue of the $p$ -Laplacian and the $p$ -biharmonic operator

Jiří Benedikt (2015)

Mathematica Bohemica

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We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet $p$ -Laplacian and the Navier $p$ -biharmonic operator on a ball of radius $R$ in $ℝ^{N}$ and its asymptotics for $p$ approaching $1$ and $\infty$ . Let $p$ tend to $\infty$ . There is a critical radius $R_{C}$ of the ball such that the principal eigenvalue goes to $\infty$ for $0 < R \leq R_{C}$ and to $0$ for $R > R_{C}$ . The critical radius is $R_{C} = 1$ for any $N \in ℕ$ for the $p$ -Laplacian and $R_{C} = \sqrt{2 N}$ in the case of the $p$ -biharmonic operator. When $p$ approaches $1$ , the principal eigenvalue...

Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space

Michael Gil' (2024)

Czechoslovak Mathematical Journal

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Let $A$ be a bounded linear operator in a complex separable Hilbert space $ℋ$ , and $S$ be a selfadjoint operator in $ℋ$ . Assuming that $A - S$ belongs to the Schatten-von Neumann ideal $𝒮_{p}$ $(p > 1),$ we derive a bound for $\sum_{k} {| R λ_{k} (A) - λ_{k} (S) |}^{p}$ , where $λ_{k} (A)$ $(k = 1, 2, \dots)$ are the eigenvalues of $A$ . Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case $p = 2$ we refine the Weyl inequality between the real parts of the eigenvalues of $A$ and the eigenvalues...

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...

Asymptotic properties of a $ϕ$ -Laplacian and Rayleigh quotient

Waldo Arriagada, Jorge Huentutripay (2020)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider the $ϕ$ -Laplacian problem with Dirichlet boundary condition, $- div (ϕ (| \nabla u |) \frac{\nabla u}{| \nabla u |}) {= λ g (\cdot) ϕ (u) in Ω, λ \in ℝ and u |}_{\partial Ω} = 0 .$ The term $ϕ$ is a real odd and increasing homeomorphism, $g$ is a nonnegative function in $L^{\infty} (Ω)$ and $Ω \subseteq ℝ^{N}$ is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both...

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 $n \geq 3$ .

On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Yusaku Yamamoto (2017)

Applications of Mathematics

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Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix $A \in ℝ^{m \times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on $Tr (A^{- 1})$ and $Tr (A^{- 2})$ have attracted attention recently, because they can be computed in $O (m)$ operations when $A$ is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that...

Partial sum of eigenvalues of random graphs

Israel Rocha (2020)

Applications of Mathematics

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Let $G$ be a graph on $n$ vertices and let $λ_{1} \geq λ_{2} \geq ... \geq λ_{n}$ be the eigenvalues of its adjacency matrix. For random graphs we investigate the sum of eigenvalues $s_{k} = \sum_{i = 1}^{k} λ_{i}$ , for $1 \leq k \leq n$ , and show that a typical graph has $s_{k} \leq (e (G) + k^{2}) / {(0.99 n)}^{1 / 2}$ , where $e (G)$ is the number of edges of $G$ . We also show bounds for the sum of eigenvalues within a given range in terms of the number of edges. The approach for the proofs was first used in Rocha (2020) to bound the partial sum of eigenvalues of the Laplacian matrix.

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 $σ (T) = σ_{a} (T)$ .

Displaying similar documents to “On the spectrum and eigenfunctions of the operator ( V f ) ( x ) = ∫ 0 x α f ( t ) d t ”

Displaying similar documents to “On the spectrum and eigenfunctions of the operator $(V f) (x) = \int_{0}^{x^{α}} f (t) d t$ ”