A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces

B. L. Chalmers; F. T. Metcalf

Displaying similar documents to “A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces”

A note on generalized projections in c₀

Beata Deręgowska, Barbara Lewandowska (2014)

Annales Polonici Mathematici

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Let V ⊂ Z be two subspaces of a Banach space X. We define the set of generalized projections by $_{V} (X, Z) : = {P \in (X, Z) : P |}_{V} = i d$ . Now let X = c₀ or $l_{\infty}^{m}$ , Z:= kerf for some f ∈ X* and $V : = Z \cap l ⁿ_{\infty}$ (n < m). The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215-227] and...

Structure of Rademacher subspaces in Cesàro type spaces

Sergey V. Astashkin, Lech Maligranda (2015)

Studia Mathematica

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The structure of the closed linear span of the Rademacher functions in the Cesàro space $C e s_{\infty}$ is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in $C e s_{\infty}$ , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of $C e s_{\infty}$ if 1 < p < ∞.

Completely 1-complemented subspaces of the Schatten spaces $S^{p}$

Jean Roydor (2007)

Banach Center Publications

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We describe the subspaces of $S^{p}$ (1 ≤ p ≠ 2 < ∞) which are the range of a completely contractive projection.

Best approximation in spaces of bounded linear operators

Grzegorz Lewicki

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CONTENTSChapter 0...............................................................................................................................................................................5 0.1. Introduction..................................................................................................................................................................5 0.2. Preliminary results.......................................................................................................................................................9Chapter...

Spaces with maximal projection constants

Hermann König, Nicole Tomczak-Jaegermann (2003)

Studia Mathematica

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We show that n-dimensional spaces with maximal projection constants exist not only as subspaces of $l_{\infty}$ but also as subspaces of l₁. They are characterized by a rigid set of vector conditions. Nevertheless, we show that, in general, there are many non-isometric spaces with maximal projection constants. Several examples are discussed in detail.

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.

On pairs of Banach spaces which are isomorphic to complemented subspaces of each other

Elói Medina Galego (2004)

Colloquium Mathematicae

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We establish the existence of Banach spaces E and F isomorphic to complemented subspaces of each other but with $E^{m} \oplus F ⁿ$ isomorphic to $E^{p} \oplus F^{q}$ , m, n, p, q ∈ ℕ, if and only if m = p and n = q.

Geometry of the Banach spaces C(βℕ × K,X) for compact metric spaces K

Dale E. Alspach, Elói Medina Galego (2011)

Studia Mathematica

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A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X). In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of $C (ω^{ω})$ then X contains a copy...

Second derivatives of norms and contractive complementation in vector-valued spaces

Bas Lemmens, Beata Randrianantoanina, Onno van Gaans (2007)

Studia Mathematica

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We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $ℓ_{p} (X)$ , where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $ℓ_{p} (X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

$L^{2}$ -Summand Vectors and Complemented Hilbertizable Subspaces

Antonio Aizpuru, Francisco J. García-Pacheco (2007)

Bollettino dell'Unione Matematica Italiana

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In this paper, we show a necessary and sufficient condition for a real Banach space to have an infinite dimensional subspace which is hilbertizable and complemented using techniques related to $L^{2}$ -summand vectors.

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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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...

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 Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2006)

Studia Mathematica

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We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type $ℒ_{\infty}$ but not conversely. Moreover, $ℒ_{\infty}$ -spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will...

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.

Distinguished subspaces of $L_{p}$ of maximal dimension

Geraldo Botelho, Daniel Cariello, Vinícius V. Fávaro, Daniel Pellegrino, Juan B. Seoane-Sepúlveda (2013)

Studia Mathematica

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The Morse minimal system is finitarily Kakutani equivalent to the binary odometer

Mrinal Kanti Roychowdhury, Daniel J. Rudolph (2008)

Fundamenta Mathematicae

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Two invertible dynamical systems (X,,μ,T) and (Y,,ν,S), where X and Y are Polish spaces and Borel probability spaces and T, S are measure preserving homeomorphisms of X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X₀ of X of measure one onto a subset Y₀ of Y of full measure such that (1) ${ϕ |}_{X ₀}$ is continuous in the relative topology on X₀ and $ϕ^{- 1} |_{Y ₀}$ is continuous in the relative topology on Y₀, (2) $ϕ (O r b_{T} (x)) = O r b_{S} (ϕ (x))$ for μ-a.e. x ∈ X. (X,,μ,T)...

Separating by $G_{δ}$ -sets in finite powers of ω₁

Yasushi Hirata, Nobuyuki Kemoto (2003)

Fundamenta Mathematicae

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It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint $G_{δ}$ -sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.

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...

On the size of quotients of function spaces on a topological group

Ahmed Bouziad, Mahmoud Filali (2011)

Studia Mathematica

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For a non-precompact topological group G, we consider the space C(G) of bounded, continuous, scalar-valued functions on G with the supremum norm, together with the subspace LMC(G) of left multiplicatively continuous functions, the subspace LUC(G) of left norm continuous functions, and the subspace WAP(G) of weakly almost periodic functions. We establish that the quotient space LUC(G)/WAP(G) contains a linear isometric copy of $ℓ_{\infty}$ , and that the quotient space C(G)/LMC(G) (and a fortiori...

Definable stratification satisfying the Whitney property with exponent 1

Beata Kocel-Cynk (2007)

Annales Polonici Mathematici

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We prove that for a finite collection of sets $A ₁, . . ., A_{s} \subset ℝ^{k + n}$ definable in an o-minimal structure there exists a compatible definable stratification such that for any stratum the fibers of its projection onto $ℝ^{k}$ satisfy the Whitney property with exponent 1.

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

Algebraic genericity of strict-order integrability

Luis Bernal-González (2010)

Studia Mathematica

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We provide sharp conditions on a measure μ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space $L^{p} (μ, X)$ (p ≥ 1) which are not q-integrable for any q > p (or any q < p) contains large subspaces of $L^{p} (μ, X)$ (without zero). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many non-q-integrable functions can even be obtained on any nonempty open subset of X, assuming that X is a topological...

A theorem on isotropic spaces

Félix Cabello Sánchez (1999)

Studia Mathematica

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Let X be a normed space and $G_{F} (X)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_{F} (X)$ acts transitively on the unit sphere then X must be an inner product space.

Powers of m-isometries

Teresa Bermúdez, Carlos Díaz Mendoza, Antonio Martinón (2012)

Studia Mathematica

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A bounded linear operator T on a Banach space X is called an (m,p)-isometry for a positive integer m and a real number p ≥ 1 if, for any vector x ∈ X, $\sum_{k = 0}^{m} {(- 1)}^{k} (\binom{m}{k}) | | T^{k} x {| |}^{p} = 0$ . We prove that any power of an (m,p)-isometry is also an (m,p)-isometry. In general the converse is not true. However, we prove that if $T^{r}$ and $T^{r + 1}$ are (m,p)-isometries for a positive integer r, then T is an (m,p)-isometry. More precisely, if $T^{r}$ is an (m,p)-isometry and $T^{s}$ is an (l,p)-isometry, then $T^{t}$ is an (h,p)-isometry, where t = gcd(r,s)...

Subspaces of $L_{p}$ , p > 2, determined by partitions and weights

Dale E. Alspach, Simei Tong (2003)

Studia Mathematica

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Many of the known complemented subspaces of $L_{p}$ have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of $L_{p}$ . It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given...

On isometrical extension properties of function spaces

Hisao Kato (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C (Q)$ and $C (Δ)$ , where $Q$ and $Δ$ denote the Hilbert cube ${[0, 1]}^{\infty}$ and a Cantor set, respectively.

On complemented copies of c₀(ω₁) in C(Kⁿ) spaces

Leandro Candido, Piotr Koszmider (2016)

Studia Mathematica

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Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes ̂_{ε}^{n} C (K)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes ̂_{ε}^{n} C (K)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X \otimes ̂_{ε} Y$ contains a complemented copy of c₀ if one of the infinite-dimensional...

The Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Studia Mathematica

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Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{Λ} (G)$ or $L ¹_{Λ} (G)$ and which quotients of the form $C (G) / C_{Λ} (G)$ or $L ¹ (G) / L ¹_{Λ} (G)$ have the Daugavet property. We show that $C_{Λ} (G)$ is a rich subspace of C(G) if and only if $Γ ∖ Λ^{- 1}$ is a semi-Riesz set. If $L ¹_{Λ} (G)$ is a rich subspace of L¹(G), then $C_{Λ} (G)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C (G) / C_{Λ} (G)$ has the Daugavet property if Λ is a Rosenthal set, and that $L ¹_{Λ} (G)$ is a poor subspace of L¹(G)...

A classification of projectors

Gustavo Corach, Alejandra Maestripieri, Demetrio Stojanoff (2005)

Banach Center Publications

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A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and $A^{1 / 2}$ . It also depends on a certain angle between A() and the orthogonal of .

Structure of Cesàro function spaces: a survey

Sergey V. Astashkin, Lech Maligranda (2014)

Banach Center Publications

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Geometric structure of Cesàro function spaces $C e s_{p} (I)$ , where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that $C e s_{p} [0, 1]$ contains isomorphic and complemented copies of $l_{q}$ -spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces $C e s_{p} [0, 1]$ .

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set $E$ with $C^{1}$ boundary and the spherical Hausdorff measure $S^{Q - 1}$ restricted to $\partial E$ , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to $S^{Q - 1} (\partial E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Decompositions for real Banach spaces with small spaces of operators

Manuel González, José M. Herrera (2007)

Studia Mathematica

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We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_{i}$ for which $(X_{i}) / ℐ n (X_{i})$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_{i}$ can be divided into subsets in such a way that if $X_{i}$ and $X_{j}$ are in different...

Three-space problems and bounded approximation properties

Wolfgang Lusky (2003)

Studia Mathematica

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Let ${R ₙ}_{n = 1}^{\infty}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_{p}$ -space, then both X and A have bases. We apply these results to show that the spaces $C_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset C ()$ and $L_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset L ₁ ()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.