Displaying similar documents to “Spectral radius of operators associated with dynamical systems in the spaces C(X)”

H calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the H calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if - A generates a bounded analytic C 0 semigroup ( T t ) on a UMD space, then the H calculus of A is bounded if and only if ( T t ) has a dilation to a bounded group on L 2 ( [ 0 , 1 ] , X ) . This generalises a Hilbert space result of C.LeMerdy. If X is an L p space we can choose another L p space in place of L 2 ( [ 0 , 1 ] , X ) .

On the Schröder equation

M. Kuczma

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CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. C p solutions in *................................................ 74. Change of variables..............................................................................

Spectral radius of weighted composition operators in L p -spaces

Krzysztof Zajkowski (2010)

Studia Mathematica

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We prove that for the spectral radius of a weighted composition operator a T α , acting in the space L p ( X , , μ ) , the following variational principle holds: l n r ( a T α ) = m a x ν M ¹ α , e X l n | a | d ν , where X is a Hausdorff compact space, α: X → X is a continuous mapping preserving a Borel measure μ with suppμ = X, M ¹ α , e is the set of all α-invariant ergodic probability measures on X, and a: X → ℝ is a continuous and -measurable function, where = n = 0 α - n ( ) . This considerably extends the range of validity of the above formula, which was previously known...

Integral equalities for functions of unbounded spectral operators in Banach spaces

Benedetto Silvestri

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The work is dedicated to investigating a limiting procedure for extending “local” integral operator equalities to “global” ones in the sense explained below, and to applying it to obtaining generalizations of the Newton-Leibniz formula for operator-valued functions for a wide class of unbounded operators. The integral equalities considered have the form g ( R F ) f x ( R F ) d μ ( x ) = h ( R F ) . (1) They involve functions of the kind X x f x ( R F ) B ( F ) , where X is a general locally compact space, F runs over a suitable class of Banach subspaces...

On open maps and related functions over the Salbany compactification

Mbekezeli Nxumalo (2024)

Archivum Mathematicum

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Given a topological space X , let 𝒰 X and η X : X 𝒰 X denote, respectively, the Salbany compactification of X and the compactification map called the Salbany map of X . For every continuous function f : X Y , there is a continuous function 𝒰 f : 𝒰 X 𝒰 Y , called the Salbany lift of f , satisfying ( 𝒰 f ) η X = η Y f . If a continuous function f : X Y has a stably compact codomain Y , then there is a Salbany extension F : 𝒰 X Y of f , not necessarily unique, such that F η X = f . In this paper, we give a condition on a space such that its Salbany map is open. In...

Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities

Boumediene Abdellaoui, Eduardo Colorado, Ireneo Peral (2004)

Journal of the European Mathematical Society

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In this work we study the problem u t div ( | x | 2 γ u ) = λ u α | x | 2 ( γ + 1 ) + f in Ω × ( 0 , T ) , u 0 in Ω × ( 0 , T ) , u = 0 on Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , Ω N ( N 2 ) is a bounded regular domain such that 0 Ω , λ > 0 , α > 0 , - < γ < ( N 2 ) / 2 , f and u 0 are positive functions such that f L 1 ( Ω × ( 0 , T ) ) and u 0 L 1 ( Ω ) . The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case α = 1 , 1 + γ > 0 ; (ii) the nonexistence of solutions if α > 1 , 1 + γ > 0 ; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions...

The boundedness of two classes of integral operators

Xin Wang, Ming-Sheng Liu (2021)

Czechoslovak Mathematical Journal

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The aim of this paper is to characterize the L p - L q boundedness of two classes of integral operators from L p ( 𝒰 , d V α ) to L q ( 𝒰 , d V β ) in terms of the parameters a , b , c , p , q and α , β , where 𝒰 is the Siegel upper half-space. The results in the presented paper generalize a corresponding result given in C. Liu, Y. Liu, P. Hu, L. Zhou (2019).

A symmetry problem in the calculus of variations

Graziano Crasta (2006)

Journal of the European Mathematical Society

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We consider the integral functional J ( u ) = Ω [ f ( | D u | ) u ] d x , u W 0 1 , 1 ( Ω ) , where Ω n , n 2 , is a nonempty bounded connected open subset of n with smooth boundary, and s f ( | s | ) is a convex, differentiable function. We prove that if J admits a minimizer in W 0 1 , 1 ( Ω ) depending only on the distance from the boundary of Ω , then Ω must be a ball.

On square functions associated to sectorial operators

Christian Le Merdy (2004)

Bulletin de la Société Mathématique de France

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We give new results on square functions x F = 0 F ( t A ) x 2 d t t 1 / 2 p associated to a sectorial operator A on L p for 1 &lt; p &lt; . Under the assumption that A is actually R -sectorial, we prove equivalences of the form K - 1 x G x F K x G for suitable functions F , G . We also show that A has a bounded H functional calculus with respect to . F . Then we apply our results to the study of conditions under which we have an estimate ( 0 | C e - t A ( x ) | 2 d t ) 1 / 2 q M x p , when - A generates a bounded semigroup e - t A on L p and C : D ( A ) L q is a linear mapping.

Linear maps preserving A -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

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Let be a complex Hilbert space, A a positive operator with closed range in ( ) and A ( ) the sub-algebra of ( ) of all A -self-adjoint operators. Assume φ : A ( ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ) = ψ ( T ) for all T A ( ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri (2003)

Colloquium Mathematicae

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We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ( P λ ) ⎩ u Ω = 0 where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ( P λ ) admits a non-zero, non-negative strong solution u λ p 2 W 2 , p ( Ω ) such that l i m λ 0 | | u λ | | W 2 , p ( Ω ) = 0 for all p ≥ 2. Moreover, the function λ I λ ( u λ ) is negative and decreasing in ]0,λ*[, where I λ is the energy functional related to ( P λ ). ...

More on exposed points and extremal points of convex sets in n and Hilbert space

Stoyu T. Barov (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let 𝕍 be a separable real Hilbert space, k with k < dim 𝕍 , and let B be convex and closed in 𝕍 . Let 𝒫 be a collection of linear k -subspaces of 𝕍 . A point w B is called exposed by 𝒫 if there is a P 𝒫 so that ( w + P ) B = { w } . We show that, under some natural conditions, B can be reconstituted as the convex hull of the closure of all its exposed by 𝒫 points whenever 𝒫 is dense and G δ . In addition, we discuss the question when the set of exposed by some 𝒫 points forms a G δ -set.

Measure-geometric Laplacians for partially atomic measures

Marc Kesseböhmer, Tony Samuel, Hendrik Weyer (2020)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the fundamental theorem of calculus, and based on the works of W. Feller as well as M. Kac and M. G. Kreĭn, given an atomless Borel probability measure η supported on a compact subset of U. Freiberg and M. Zähle introduced a measure-geometric approach to define a first order differential operator η and a second order differential operator Δ η , with respect to η . We generalize this approach to measures of the form η : = ν + δ , where ν is non-atomic and δ is finitely supported. We determine...

Renormings of c 0 and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this paper is to show that for every Banach space ( X , · ) containing asymptotically isometric copy of the space c 0 there is a bounded, closed and convex set C X with the Chebyshev radius r ( C ) = 1 such that for every k 1 there exists a k -contractive mapping T : C C with x - T x > 1 1 / k for any x C .

Truncated spectral regularization for an ill-posed non-linear parabolic problem

Ajoy Jana, M. Thamban Nair (2019)

Czechoslovak Mathematical Journal

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It is known that the nonlinear nonhomogeneous backward Cauchy problem u t ( t ) + A u ( t ) = f ( t , u ( t ) ) , 0 t < τ with u ( τ ) = φ , where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f , that a solution of the above problem satisfies an integral equation involving the spectral representation of A , which is also ill-posed. Spectral truncation...

The weak Gelfand-Phillips property in spaces of compact operators

Ioana Ghenciu (2017)

Commentationes Mathematicae Universitatis Carolinae

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For Banach spaces X and Y , let K w * ( X * , Y ) denote the space of all w * - w continuous compact operators from X * to Y endowed with the operator norm. A Banach space X has the w G P property if every Grothendieck subset of X is relatively weakly compact. In this paper we study Banach spaces with property w G P . We investigate whether the spaces K w * ( X * , Y ) and X ϵ Y have the w G P property, when X and Y have the w G P property.