On the functional equation defined by Lie's product formula

Gerd Herzog; Christoph Schmoeger

Displaying similar documents to “On the functional equation defined by Lie's product formula”

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

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This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form $F (x) : = F ̃ x^{(2 n + 1)}$ where $F ̃ : X^{2 n + 1} \to Y$ is a continuous (2n+1)-linear operator.

The functional equation $f^{2} (x) = g (x)$

James C. Lillo (1967)

Annales Polonici Mathematici

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Continuous solutions of the functional equation $φ (f (x)) = G (x, φ (x))$ for vector-valued functions φ

Z. Krzeszowiak (1969)

Annales Polonici Mathematici

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(Non-)amenability of ℬ(E)

Volker Runde (2010)

Banach Center Publications

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In 1972, the late B. E. Johnson introduced the notion of an amenable Banach algebra and asked whether the Banach algebra ℬ(E) of all bounded linear operators on a Banach space E could ever be amenable if dim E = ∞. Somewhat surprisingly, this question was answered positively only very recently as a by-product of the Argyros-Haydon result that solves the “scalar plus compact problem”: there is an infinite-dimensional Banach space E, the dual of which is ℓ¹, such that $ℬ (E) = (E) + ℂ i d_{E}$ . Still, ℬ(ℓ²) is...

On Some Properties of Separately Increasing Functions from [0,1]ⁿ into a Banach Space

Artur Michalak (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f : {[0, 1]}^{m} \to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f : {[0, 1]}^{m} \to X$ with respect to any norming subset there exists a separately increasing function $g : {[0, 1]}^{m} \to ℝ$ such that the sets of...

On some properties of generalized Marcinkiewicz spaces

Evgeniy Pustylnik (2001)

Studia Mathematica

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We give a full solution of the following problems concerning the spaces $M_{φ} (X ⃗)$ : (i) to what extent two functions φ and ψ should be different in order to ensure that $M_{φ} (X ⃗) \neq M_{ψ} (X ⃗)$ for any nontrivial Banach couple X⃗; (ii) when an embedding $M_{φ} (X ⃗) ⊊ M_{ψ} (X ⃗)$ can (or cannot) be dense; (iii) which Banach space can be regarded as an $M_{φ} (X ⃗)$ -space for some (unknown beforehand) Banach couple X⃗.

Geometry of Banach spaces and biorthogonal systems

S. Dilworth, Maria Girardi, W. Johnson (2000)

Studia Mathematica

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A separable Banach space X contains $ℓ_{1}$ isomorphically if and only if X has a bounded fundamental total $w c_{0} *$ -stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total $w c_{0} *$ -biorthogonal system.

On the existence of non-linear frames

Shah Jahan, Varinder Kumar, S.K. Kaushik (2017)

Archivum Mathematicum

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A stronger version of the notion of frame in Banach space called Strong Retro Banach frame (SRBF) is defined and studied. It has been proved that if $𝒳$ is a Banach space such that $𝒳^{*}$ has a SRBF, then $𝒳$ has a Bi-Banach frame with some geometric property. Also, it has been proved that if a Banach space $𝒳$ has an approximative Schauder frame, then $𝒳^{*}$ has a SRBF. Finally, the existence of a non-linear SRBF in the conjugate of a separable Banach space has been proved.

Spaces of operators and c₀

P. Lewis (2001)

Studia Mathematica

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Bessaga and Pełczyński showed that if c₀ embeds in the dual X* of a Banach space X, then ℓ¹ embeds complementably in X, and $ℓ^{\infty}$ embeds as a subspace of X*. In this note the Diestel-Faires theorem and techniques of Kalton are used to show that if X is an infinite-dimensional Banach space, Y is an arbitrary Banach space, and c₀ embeds in L(X,Y), then $ℓ^{\infty}$ embeds in L(X,Y), and ℓ¹ embeds complementably in $X \otimes_{γ} Y *$ . Applications to embeddings of c₀ in various spaces of operators are given.

$C^{r}$ -solutions of a system of functional equations

Z. Kominek (1974)

Annales Polonici Mathematici

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On the functional equation $f (x) + \sum_{i = 1}^{n} g_{i} (y_{i}) = h (T (x, y_{1}, y_{2}, . . ., y_{n}))$

C. T. Ng (1973)

Annales Polonici Mathematici

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On the solutions of the functional equation $ϕ (x) + ϕ^{2} (x) = F (x)$

M. Malenica (1982)

Matematički Vesnik

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On some symmetrical equations of the form $f_{1} (x_{1} + . . . + x_{n}) = \sum_{i_{1}, . . ., i_{n}}) f_{1} (x_{i} 1) . . . f_{n} (x_{i} n)$

H. Światak (1967)

Annales Polonici Mathematici

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Reflexivity and approximate fixed points

Eva Matoušková, Simeon Reich (2003)

Studia Mathematica

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A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $x_{n_{k}}$ for which $s u p_{f \in S_{X *}} l i m s u p_{k \to \infty} f (x_{n_{k}})$ is attained at some f in the dual unit sphere $S_{X *}$ . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every $f \in S_{X *}$ , there exists $g \in S_{X *}$ such that $l i m s u p_{n \to \infty} f (x ₙ) < l i m i n f_{n \to \infty} g (x ₙ)$ . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded...

$L_{1}$ contains every two-dimensional normed space

David Yost (1988)

Annales Polonici Mathematici

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Multiplying balls in the space of continuous functions on [0,1]

Marek Balcerzak, Artur Wachowicz, Władysław Wilczyński (2005)

Studia Mathematica

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Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set $Φ^{- 1} (E)$ is residual whenever E is...

Linearization of isometric embedding on Banach spaces

Yu Zhou, Zihou Zhang, Chunyan Liu (2015)

Studia Mathematica

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Let X,Y be Banach spaces, f: X → Y be an isometry with f(0) = 0, and $T : \bar{s p a n} (f (X)) \to X$ be the Figiel operator with $T \circ f = I d_{X}$ and ||T|| = 1. We present a sufficient and necessary condition for the Figiel operator T to admit a linear isometric right inverse. We also prove that such a right inverse exists when $\bar{s p a n} (f (X))$ is weakly nearly strictly convex.

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$ , belonging to the class $L^{p}(J,E)$ , $p>1$ . Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

Estimation of the Szlenk index of Banach spaces via Schreier spaces

Ryan Causey (2013)

Studia Mathematica

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For each ordinal α < ω₁, we prove the existence of a Banach space with a basis and Szlenk index $ω^{α + 1}$ which is universal for the class of separable Banach spaces with Szlenk index not exceeding $ω^{α}$ . Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.

On monotonic functions from the unit interval into a Banach space with uncountable sets of points of discontinuity

Artur Michalak (2003)

Studia Mathematica

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We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. We show that if f: [0,1] → X is an increasing function with respect to a norming subset E of X* with uncountably many points of discontinuity and Q is a countable dense subset of [0,1], then (1) $\bar{l i n f ([0, 1])}$ contains an order isomorphic copy of D(0,1), (2) $\bar{l i n f (Q)}$ contains an isomorphic copy of C([0,1]), (3) $\bar{l i n f ([0, 1])} / \bar{l i n f (Q)}$ contains an isomorphic copy of c₀(Γ) for some uncountable...

Similarity-preserving linear maps on B(X)

Fangyan Lu, Chaoran Peng (2012)

Studia Mathematica

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Let X be an infinite-dimensional Banach space, and B(X) the algebra of all bounded linear operators on X. Then ϕ: B(X) → B(X) is a bijective similarity-preserving linear map if and only if one of the following holds: (1) There exist a nonzero complex number c, an invertible bounded operator T in B(X) and a similarity-invariant linear functional h on B(X) with h(I) ≠ -c such that $ϕ (A) = c T A T^{- 1} + h (A) I$ for all A ∈ B(X). (2) There exist a nonzero complex number c, an invertible bounded linear operator T: X*...

On the mutually non isomorphic $l_{p} (l_{q})$

Pilar Cembranos, Jose Mendoza (2016)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this note we survey the partial results needed to show the following general theorem: $l_{p} (l_{q}) : 1 \leq p, q \leq + \infty$ is a family of mutually non isomorphic Banach spaces. We also comment some related facts and open problems.

Method of averaging for the system of functional-differential inclusions

Teresa Janiak, Elżbieta Łuczak-Kumorek (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The basic idea of this paper is to give the existence theorem and the method of averaging for the system of functional-differential inclusions of the form ⎧ $ẋ (t) \in F (t, x_{t}, y_{t})$ (0) ⎨ ⎩ $ẋ (t) \in G (t, x_{t}, y_{t})$ (1)

Solutions for the p-order Feigenbaum’s functional equation $h (g (x)) = g^{p} (h (x))$

Min Zhang, Jianguo Si (2014)

Annales Polonici Mathematici

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This work deals with Feigenbaum’s functional equation ⎧ $h (g (x)) = g^{p} (h (x))$ , ⎨ ⎩ g(0) = 1, -1 ≤ g(x) ≤ 1, x∈[-1,1] where p ≥ 2 is an integer, $g^{p}$ is the p-fold iteration of g, and h is a strictly monotone odd continuous function on [-1,1] with h(0) = 0 and |h(x)| < |x| (x ∈ [-1,1], x ≠ 0). Using a constructive method, we discuss the existence of continuous unimodal even solutions of the above equation.

The extension of the Krein-Šmulian theorem for order-continuous Banach lattices

Antonio S. Granero, Marcos Sánchez (2008)

Banach Center Publications

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If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) $d ̂ ({\bar{c o}}^{w *} (K), X) \leq 2 d ̂ (K, X)$ and, if K ∩ X is w*-dense in K, then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ ; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ ; (iii) if X has a 1-symmetric basis, then $d ̂ ({\bar{c o}}^{w *} (K), X) = d ̂ (K, X)$ .

Rosenthal operator spaces

M. Junge, N. J. Nielsen, T. Oikhberg (2008)

Studia Mathematica

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In 1969 Lindenstrauss and Rosenthal showed that if a Banach space is isomorphic to a complemented subspace of an $L_{p}$ -space, then it is either an $L_{p}$ -space or isomorphic to a Hilbert space. This is the motivation of this paper where we study non-Hilbertian complemented operator subspaces of non-commutative $L_{p}$ -spaces and show that this class is much richer than in the commutative case. We investigate the local properties of some new classes of operator spaces for every 2 < p < ∞ which...

On the Aronszajn property for integral equations in Banach space

Stanisław Szufla (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Almost multiplicative functions on commutative Banach algebras

S. H. Kulkarni, D. Sukumar (2010)

Studia Mathematica

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Let A be a complex commutative Banach algebra with unit 1 and δ > 0. A linear map ϕ: A → ℂ is said to be δ-almost multiplicative if |ϕ(ab) - ϕ(a)ϕ(b)| ≤ δ||a|| ||b|| for all a,b ∈ A. Let 0 < ϵ < 1. The ϵ-condition spectrum of an element a in A is defined by $σ_{ϵ} (a) : = λ \in ℂ : | | λ - {a | | | | (λ - a)}^{- 1} | | \geq 1 / ϵ$ with the convention that $| | λ - {a | | | | (λ - a)}^{- 1} | | = \infty$ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If ϕ(1) = 1 and ϕ is δ-almost multiplicative, then $ϕ (a) \in σ_{δ} (a)$ for all a in A. (2) If ϕ is linear and $ϕ (a) \in σ_{ϵ} (a)$ for...

The Dual of a Non-reflexive L-embedded Banach Space Contains $l^{\infty}$ Isometrically

Hermann Pfitzner (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

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A Banach space is said to be L-embedded if it is complemented in its bidual in such a way that the norm between the two complementary subspaces is additive. We prove that the dual of a non-reflexive L-embedded Banach space contains $l^{\infty}$ isometrically.

On the functional equation $f (x + y - x y) + f (x y) = f (x) + f (y)$

H. Swiatak (1968)

Matematički Vesnik

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On the number of non-isomorphic subspaces of a Banach space

Valentin Ferenczi, Christian Rosendal (2005)

Studia Mathematica

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We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ${(e_{i})}_{i \in ℕ}$ ; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, ${[e_{i}]}_{i \in A} ≇ {[e_{i}]}_{i \in B}$ , or for a residual set of infinite subsets A of ℕ, ${[e_{i}]}_{i \in A}$ is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to $\oplus {[e_{i}]}_{i \in D}$ for any D ⊂ ℕ, and isomorphic to a denumerable Schauder...

The law of large numbers and a functional equation

Maciej Sablik (1998)

Annales Polonici Mathematici

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We deal with the linear functional equation (E) $g (x) = \sum_{i = 1}^{r} p_{i} g (c_{i} x)$ , where g:(0,∞) → (0,∞) is unknown, $(p ₁, . . ., p_{r})$ is a probability distribution, and $c_{i}$ ’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

Isometries between groups of invertible elements in Banach algebras

Osamu Hatori (2009)

Studia Mathematica

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We show that if T is an isometry (as metric spaces) from an open subgroup of the group of invertible elements in a unital semisimple commutative Banach algebra A onto a open subgroup of the group of invertible elements in a unital Banach algebra B, then $T {(1)}^{- 1} T$ is an isometrical group isomorphism. In particular, $T {(1)}^{- 1} T$ extends to an isometrical real algebra isomorphism from A onto B.

Shift-invariant functionals on Banach sequence spaces

Albrecht Pietsch (2013)

Studia Mathematica

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The present paper is a continuation of [23], from which we know that the theory of traces on the Marcinkiewicz operator ideal $(H) : = T \in (H) : s u p_{1 \leq m < \infty} 1 / (l o g m + 1) \sum_{n = 1}^{m} a ₙ (T) < \infty$ can be reduced to the theory of shift-invariant functionals on the Banach sequence space $(ℕ ₀) : = c = (γ_{l}) : s u p_{0 \leq k < \infty} 1 / (k + 1) \sum_{l = 0}^{k} | γ_{l} | < \infty$ . The final purpose of my studies, which will be finished in [24], is the following. Using the density character as a measure, I want to determine the size of some subspaces of the dual *(H). Of particular interest are the sets formed by the Dixmier traces and the Connes-Dixmier...

Pisier's inequality revisited

Tuomas Hytönen, Assaf Naor (2013)

Studia Mathematica

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Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies $\int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} \partial_{j} f_{j} {(ε) | |}^{p} d μ (ε) \leq^{p} \int_{- 1, 1 ⁿ} \int_{- 1, 1 ⁿ} | | \sum_{j = 1}^{n} δ_{j} Δ f_{j} (ε) {| |}^{p} d μ (ε) d μ (δ)$ , where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${\partial_{j}}_{j = 1}^{n}$ and $Δ = \sum_{j = 1}^{n} \partial_{j}$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $ⁿ_{p} (X)$ , we show that $ⁿ_{p} (X) \leq \sum_{k = 1}^{n} 1 / k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...