Perturbation of analytic operators and temporal regularity of discrete heat kernels

Sönke Blunck

Displaying similar documents to “Perturbation of analytic operators and temporal regularity of discrete heat kernels”

Co-analytic, right-invertible operators are supercyclic

Sameer Chavan (2010)

Colloquium Mathematicae

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Let denote a complex, infinite-dimensional, separable Hilbert space, and for any such Hilbert space , let () denote the algebra of bounded linear operators on . We show that for any co-analytic, right-invertible T in (), αT is hypercyclic for every complex α with $| α | > β^{- 1}$ , where $β \equiv i n f_{| | x | | = 1} | | T * x | | > 0$ . In particular, every co-analytic, right-invertible T in () is supercyclic.

Overstability and resonance

Augustin Fruchard, Reinhard Schäfke (2003)

Annales de l’institut Fourier

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We consider a singularity perturbed nonlinear differential equation $ε u^{'} = f (x) u + + ε P (x, u, ε)$ which we suppose real analytic for $x$ near some interval $[a, b]$ and small $| u |$ , $| ε |$ . We furthermore suppose that 0 is a turning point, namely that $x f (x)$ is positive if $x \neq 0$ . We prove that the existence of nicely behaved (as $ϵ \to 0$ ) local (at $x = 0$ ) or global, real analytic or $C^{\infty}$ solutions is equivalent to the existence of a formal series solution $\sum u_{n} (x) ε^{n}$ with $u_{n}$ analytic at $x = 0$ . The main tool of a proof is a new “principle of analytic continuation” for...

The analytic $α$ -capacity and the Cauchy integral

K. Adžievski (1986)

Matematički Vesnik

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Solutions of non-homogeneous linear differential equations in the unit disc

Ting-Bin Cao, Zhong-Shu Deng (2010)

Annales Polonici Mathematici

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The main purpose of this paper is to consider the analytic solutions of the non-homogeneous linear differential equation $f^{(k)} + a_{k - 1} (z) f^{(k - 1)} + \dots + a ₁ (z) f^{'} + a ₀ (z) f = F (z)$ , where all coefficients $a ₀, a ₁, . . ., a_{k - 1}$ , F ≢ 0 are analytic functions in the unit disc = z∈ℂ: |z|<1. We obtain some results classifying the growth of finite iterated order solutions in terms of the coefficients with finite iterated type. The convergence exponents of zeros and fixed points of solutions are also investigated.

Square functions, bounded analytic semigroups, and applications

Christian Le Merdy (2007)

Banach Center Publications

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To any bounded analytic semigroup on Hilbert space or on $L^{p}$ -space, one may associate natural ’square functions’. In this survey paper, we review old and recent results on these square functions, as well as some extensions to various classes of Banach spaces, including noncommutative $L^{p}$ -spaces, Banach lattices, and their subspaces. We give some applications to $H^{\infty}$ functional calculus, similarity problems, multiplier theory, and control theory.

$^{\infty}$ -vectors and boundedness

Jan Stochel, F. H. Szafraniec (1997)

Annales Polonici Mathematici

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The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its $^{\infty}$ -vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.

Radii of p-valence of certain analytic functions of order $α$ with negative coefficients

M. K. Aouf (1989)

Matematički Vesnik

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Separately analytic functions and envelopes of holomorphy of some lower dimensional subsets of $C^{n}$

J. Siciak (1969)

Annales Polonici Mathematici

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Persistence of fixed points under rigid perturbations of maps

Salvador Addas-Zanata, Pedro A. S. Salomão (2014)

Fundamenta Mathematicae

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Let f: S¹ × [0,1] → S¹ × [0,1] be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift f̃: ℝ × [0,1] → ℝ × [0,1] we have Fix(f̃) = ℝ × 0 and that f̃ positively translates points in ℝ × 1. Let $f ̃_{ϵ}$ be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that $F i x (f ̃_{ϵ}) = \emptyset$ for all ϵ > 0 sufficiently small. The proof follows from Kerékjártó’s construction of Brouwer lines for orientation preserving...

Algebraic and analytic properties of solutions of abstract differential equations

R. Bittner

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CONTENTSINTRODUCTION............................................................................................................................... 3Chapter I. ALGEBRAIC PROPERTIES OF SOLUTIONS OF ABSTRACT DIFFERENTIALEQUATIONS§ 1. Ordinary abstract differential equations1. Taylor’s formula for an abstract derivative.......................................................................... 42 π-solutions....................................................................................................................................

Analytic semigroups on vector valued noncommutative $L^{p}$ -spaces

Cédric Arhancet (2013)

Studia Mathematica

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We give sufficient conditions on an operator space E and on a semigroup of operators on a von Neumann algebra M to obtain a bounded analytic or R-analytic semigroup ( ${(T \otimes I d_{E})}_{t \geq 0}$ on the vector valued noncommutative $L^{p}$ -space $L^{p} (M, E)$ . Moreover, we give applications to the $H^{\infty} (Σ_{θ})$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.

Composition operators: $N_{α}$ to the Bloch space to $Q_{β}$

Jie Xiao (2000)

Studia Mathematica

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Let $N_{α}$ ,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_{β}$ are Möbius invariant, but $N_{α}$ is not. We characterize, in function-theoretic terms, when the composition operator $C_{ϕ} f = f ◦ ϕ$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_{ϕ} : N_{α} \to B$ , $B \to Q_{β}$ , $N_{α} \to Q_{β}$ which is bounded resp. compact.

The Lindelöf property and σ-fragmentability

B. Cascales, I. Namioka (2003)

Fundamenta Mathematicae

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In the previous paper, we, together with J. Orihuela, showed that a compact subset X of the product space ${[- 1, 1]}^{D}$ is fragmented by the uniform metric if and only if X is Lindelöf with respect to the topology γ(D) of uniform convergence on countable subsets of D. In the present paper we generalize the previous result to the case where X is K-analytic. Stated more precisely, a K-analytic subspace X of ${[- 1, 1]}^{D}$ is σ-fragmented by the uniform metric if and only if (X,γ(D)) is Lindelöf, and if this is...

Rings of PDE-preserving operators on nuclearly entire functions

Henrik Petersson (2004)

Studia Mathematica

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Let E,F be Banach spaces where F = E’ or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, $ℋ_{N b} (E)$ , and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on $ℋ_{N b} (E)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $φ (D) \equiv^{t} φ$ , φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on $ℋ_{N b} (E)$ is PDE-preserving for a set ℙ ⊆...

Bounded point evaluations for multicyclic operators

M. EL Guendafi, M. Mbekhta, E. H. Zerouali (2005)

Banach Center Publications

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Let T be a multicyclic operator defined on some Banach space. Bounded point evaluations and analytic bounded point evaluations for T are defined to generalize the cyclic case. We extend some known results on cyclic operators to the more general setting of multicyclic operators on Banach spaces. In particular we show that if T satisfies Bishop’s property (β), then $ℬ_{a} = ℬ ∖ σ_{a p} (T)$ . We introduce the concept of analytic structures and we link it to different spectral quantities. We apply this concept...

Uniform analytic-Gevrey regularity of solutions to a semilinear heat equation

Todor Gramchev, Grzegorz Łysik (2008)

Banach Center Publications

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We study the Gevrey regularity down to t = 0 of solutions to the initial value problem for a semilinear heat equation $\partial_{t} u - Δ u = u^{M}$ . The approach is based on suitable iterative fixed point methods in $L^{p}$ based Banach spaces with anisotropic Gevrey norms with respect to the time and the space variables. We also construct explicit solutions uniformly analytic in t ≥ 0 and x ∈ ℝⁿ for some conservative nonlinear terms with symmetries.

Inclusion and neighborhood properties of certain subclasses of p-valent functions of complex order defined by convolution

R. M. El-Ashwah, M. K. Aouf, S. M. El-Deeb (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we introduce and investigate three new subclasses of $p$ -valent analytic functions by using the linear operator $D_{λ, p}^{m} (f * g) (z)$ . The various results obtained here for each of these function classes include coefficient bounds, distortion inequalities and associated inclusion relations for $(n, θ)$ -neighborhoods of subclasses of analytic and multivalent functions with negative coefficients, which are defined by means of a non-homogenous differential equation.

Heat kernel estimates for a class of higher order operators on Lie groups

Nick Dungey (2005)

Studia Mathematica

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Let G be a Lie group of polynomial volume growth. Consider a differential operator H of order 2m on G which is a sum of even powers of a generating list $A ₁, . . ., A_{d^{'}}$ of right invariant vector fields. When G is solvable, we obtain an algebraic condition on the list $A ₁, . . ., A_{d^{'}}$ which is sufficient to ensure that the semigroup kernel of H satisfies global Gaussian estimates for all times. For G not necessarily solvable, we state an analytic condition on the list which is necessary and sufficient for global...

Operator theoretic properties of semigroups in terms of their generators

S. Blunck, L. Weis (2001)

Studia Mathematica

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Let $(T_{t})$ be a C₀ semigroup with generator A on a Banach space X and let be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to if and only if the integrated semigroup $S_{t} : = \int_{0}^{t} T_{s} d s$ belongs to . For analytic semigroups, $S_{t} \in$ implies $T_{t} \in$ , and in this case we give precise estimates for the growth of the -norm of $T_{t}$ (e.g. the trace of $T_{t}$ ) in terms of the resolvent growth and the imbedding D(A) ↪ X.

A Hörmander-type spectral multiplier theorem for operators without heat kernel

Sönke Blunck (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Hörmander’s famous Fourier multiplier theorem ensures the $L_{p}$ -boundedness of $F (- Δ_{ℝ} D)$ whenever $F \in ℋ (s)$ for some $s > \frac{D}{2}$ , where we denote by $ℋ (s)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A \geq 0$ and yield the $L_{p}$ -boundedness of $F (A)$ provided $F \in ℋ (s)$ for some $s$ sufficiently large. The harmonic oscillator $A = - Δ_{ℝ} + x^{2}$ shows that in general $s > \frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In...