Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

Małgorzata Marta Czerwińska; Anna Kamińska

Displaying similar documents to “Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators”

Extreme symmetric norms on $R^{2}$

Ryszard Grząślewicz (1988)

Colloquium Mathematicae

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Extreme operators on 2-dimensional $l_{p}$ -spaces

Ryszard Grząślewicz (1981)

Colloquium Mathematicae

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Extreme points of the complex binary trilinear ball

Fernando Cobos, Thomas Kühn, Jaak Peetre (2000)

Studia Mathematica

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We characterize all the extreme points of the unit ball in the space of trilinear forms on the Hilbert space $ℂ^{2}$ . This answers a question posed by R. Grząślewicz and K. John [7], who solved the corresponding problem for the real Hilbert space $ℝ^{2}$ . As an application we determine the best constant in the inequality between the Hilbert-Schmidt norm and the norm of trilinear forms.

On some global and local geometric properties of Calderón-Lozanovskiĭ spaces

Agata Narloch (2005)

Banach Center Publications

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Criteria for full k-rotundity (k ∈ ℕ, k ≥ 2) and uniform rotundity in every direction of Calderón-Lozanovskiĭ spaces are formulated. A characterization of $H_{μ}$ -points in these spaces is also given.

Complex Convexity of Orlicz-Lorentz Spaces and its Applications

Changsun Choi, Anna Kamińska, Han Ju Lee (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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We give sufficient and necessary conditions for complex extreme points of the unit ball of Orlicz-Lorentz spaces, as well as we find criteria for the complex rotundity and uniform complex rotundity of these spaces. As an application we show that the set of norm-attaining operators is dense in the space of bounded linear operators from $d *_{(} w, 1)$ into d(w,1), where $d *_{(} w, 1)$ is a predual of a complex Lorentz sequence space d(w,1), if and only if wi ∈ c₀∖ℓ₂.

Scalar differential concomitants of first order of a symmetric connexion $Γ_{j k}^{i}$ in the two-dimensional space and their applications

Michał Lorens (1980)

Annales Polonici Mathematici

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Smooth operators in the commutant of a contraction

Pascale Vitse (2003)

Studia Mathematica

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For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the $H^{\infty}$ calculus, $H^{\infty} (T)$ , and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes $S_{p}$ .

The generalized Day norm. Part I. Properties

Monika Budzyńska, Aleksandra Grzesik, Mariola Kot (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we introduce a modification of the Day norm in $c_{0} (Γ)$ and investigate properties of this norm.

Subharmonicity in von Neumann algebras

Thomas Ransford, Michel Valley (2005)

Studia Mathematica

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Let ℳ be a von Neumann algebra with unit $1_{ℳ}$ . Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by $μ_{t} {(x)}_{t \geq 0}$ the generalized s-numbers of x, defined by $μ_{t} (x)$ = inf||xe||: e is a projection in ℳ i with $τ (1_{ℳ} - e)$ ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, $λ \mapsto \int_{0}^{t} l o g μ_{s} (f (λ)) d s$ is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.

Characterization of Strongly Exposed Points in General Köthe-Bochner Banach Spaces

Houcine Benabdellah, My Hachem Lalaoui Rhali (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study strongly exposed points in general Köthe-Bochner Banach spaces X(E). We first give a characterization of strongly exposed points of the set of X-selections of a measurable multifunction Γ. We then apply this result to the study of strongly exposed points of the closed unit ball of X(E). Precisely we show that if an element f is a strongly exposed point of $B_{X (E)}$ , then |f| is a strongly exposed point of $B_{X}$ and f(ω)/∥ f(ω)∥ is a strongly exposed point of $B_{E}$ for μ-almost all ω ∈ S(f). ...

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with $s u p ₙ | | T ⁿ / n^{w} | | < \infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{- 1} \sum_{k = 0}^{n - 1} T^{k}$ converges uniformly; (ii) $c l (I - T) X = z \in X : l i m_{n} \sum_{k = 1}^{n} T^{k} z / k e x i s t s$ .

Modulus of dentability in $L ¹ + L^{\infty}$

Adam Bohonos, Ryszard Płuciennik (2008)

Banach Center Publications

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We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L ¹ + L^{\infty} .$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L ¹ + L^{\infty}$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L ¹ + L^{\infty}$ is empty.

On some ergodic properties for continuous and affine functions

Charles J. K. Batty (1978)

Annales de l'institut Fourier

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Two problems posed by Choquet and Foias are solved: (i) Let $T$ be a positive linear operator on the space $C (X)$ of continuous real-valued functions on a compact Hausdorff space $X$ . It is shown that if $n^{- 1} \sum_{r = 0}^{n - 1} T^{r} 1$ converges pointwise to a continuous limit, then the convergence is uniform on $X$ . (ii) An example is given of a Choquet simplex $K$ and a positive linear operator $T$ on the space $A (K)$ of continuous affine real-valued functions on $K$ , such that $\inf {(T^{n} 1) (x) : n \geq} < 1$ for each...

Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case

Ilona Królak (2006)

Banach Center Publications

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We study a certain class of von Neumann algebras generated by selfadjoint elements $ω_{i} = a_{i} + a ⁺_{i}$ , where $a_{i}, a ⁺_{i}$ satisfy the general commutation relations: $a_{i} a ⁺_{j} = \sum_{r, s} t_{j}^{i} r_{s} a ⁺_{r} a_{s} + δ_{i j} I d$ . We assume that the operator T for which the constants $t_{j}^{i} r_{s}$ are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality in the case of infinite dimensional underlying space were shown. In this paper we prove that under certain conditions on the...

Multiple solutions to a perturbed Neumann problem

Giuseppe Cordaro (2007)

Studia Mathematica

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We consider the perturbed Neumann problem ⎧ -Δu + α(x)u = α(x)f(u) + λg(x,u) a.e. in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω is an open bounded set in $ℝ^{N}$ with boundary of class C², $α \in L^{\infty} (Ω)$ with $e s s i n f_{Ω} α > 0$ , f: ℝ → ℝ is a continuous function and g: Ω × ℝ → ℝ, besides being a Carathéodory function, is such that, for some p > N, $s u p_{| s | \leq t} | g (\cdot, s) | \in L^{p} (Ω)$ and $g (\cdot, t) \in L^{\infty} (Ω)$ for all t ∈ ℝ. In this setting, supposing only that the set of global minima of the function $1 / 2 ξ ² - \int_{0}^{ξ} f (t) d t$ has M ≥ 2 bounded connected components, we prove that, for all λ ∈ ℝ small enough,...

On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces

Mikio Kato, Lech Maligranda, Yasuji Takahashi (2001)

Studia Mathematica

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Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant $C_{N J} (X)$ , and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between $C_{N J} (X)$ and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the $C_{N J} (X)$ -constant, which implies that a Banach space with $C_{N J} (X)$ -constant less than 5/4 has the fixed point property. ...

Essential norms of the Neumann operator of the arithmetical mean

Josef Král, Dagmar Medková (2001)

Mathematica Bohemica

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Let $K \subset ℝ^{m}$ ( $m \geq 2$ ) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let $C_{0}^{(1)}$ be the class of all continuously differentiable real-valued functions with compact support in $ℝ^{m}$ and denote by $σ_{m}$ the area of the unit sphere in $ℝ^{m}$ . With each $ϕ \in C_{0}^{(1)}$ we associate the function $W_{K} ϕ (z) = \frac{1}{σ_{m}} \underset{ℝ^{m} ∖ K}{\to} \int g r a d ϕ (x) \cdot \frac{z - x}{{| z - x |}^{m}} x$ of the variable $z \in K$ (which is continuous in $K$ and harmonic in $K ∖ B$ ). $W_{K} ϕ$ depends only on the restriction ${ϕ |}_{B}$ of $ϕ$ to the boundary $B$ of $K$ . This gives rise to a linear operator $W_{K}$ ...

Infinitely many positive solutions for the Neumann problem involving the p-Laplacian

Giovanni Anello, Giuseppe Cordaro (2003)

Colloquium Mathematicae

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We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $- Δ_{p} u + λ (x) {| u |}^{p - 2} u = μ f (x, u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $λ \in L^{\infty} (Ω)$ with $e s s i n f_{x \in Ω} λ (x) > 0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

Extremal sections of complex $l_{p}$ -balls, 0 < p ≤ 2

Alexander Koldobsky, Marisa Zymonopoulou (2003)

Studia Mathematica

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We study the extremal volume of central hyperplane sections of complex n-dimensional $l_{p}$ -balls with 0 < p ≤ 2. We show that the minimum corresponds to hyperplanes orthogonal to vectors ξ = (ξ¹,...,ξⁿ) ∈ ℂⁿ with |ξ¹| = ... = |ξⁿ|, and the maximum corresponds to hyperplanes orthogonal to vectors with only one non-zero coordinate.

The transfinite diameter of the real ball and simplex

T. Bloom, L. Bos, N. Levenberg (2012)

Annales Polonici Mathematici

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We calculate the transfinite diameter for the real unit ball $B_{d} : = x \in ℝ^{d} : | x | \leq 1$ and the real unit simplex $T_{d} : = x \in ℝ_{+}^{d} : \sum_{j = 1}^{d} x_{j} \leq 1 .$

Symmetric products of the Euclidean spaces and the spheres

Naotsugu Chinen (2015)

Commentationes Mathematicae Universitatis Carolinae

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By $F_{n} (X)$ , $n \geq 1$ , we denote the $n$ -th symmetric product of a metric space $(X, d)$ as the space of the non-empty finite subsets of $X$ with at most $n$ elements endowed with the Hausdorff metric $d_{H}$ . In this paper we shall describe that every isometry from the $n$ -th symmetric product $F_{n} (X)$ into itself is induced by some isometry from $X$ into itself, where $X$ is either the Euclidean space or the sphere with the usual metrics. Moreover, we study the $n$ -th symmetric product of the Euclidean space up to bi-Lipschitz equivalence...