scholarly journals A BILINEAR RUBIO DE FRANCIA INEQUALITY FOR ARBITRARY SQUARES

2016 ◽  
Vol 4 ◽  
Author(s):  
CRISTINA BENEA ◽  
FRÉDÉRIC BERNICOT
Keyword(s):  
Banach Spaces ◽  
Rubio De Francia Inequality ◽  
Arbitrary Collection ◽  
Image Position ◽  
Prime End

We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane $$\begin{eqnarray}(f,g)\mapsto \biggl(\mathop{\sum }_{\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}}\biggl|\int _{\mathbb{R}^{2}}\hat{f}(\unicode[STIX]{x1D709}){\hat{g}}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})e^{2\unicode[STIX]{x1D70B}ix(\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702})}\,d\unicode[STIX]{x1D709}\,d\unicode[STIX]{x1D702}\biggr|^{r}\biggr)^{1/r},\end{eqnarray}$$ provided $r>2$. More exactly, we show that the above operator maps $L^{p}\times L^{q}\rightarrow L^{s}$ whenever $p,q,s^{\prime }$ are in the ‘local $L^{r^{\prime }}$’ range, that is, $$\begin{eqnarray}\frac{1}{p}+\frac{1}{q}+\frac{1}{s^{\prime }}=1,\quad 0\leqslant \frac{1}{p},\frac{1}{q}<\frac{1}{r^{\prime }},\quad \text{and}\quad \frac{1}{s^{\prime }}<\frac{1}{r^{\prime }}.\end{eqnarray}$$ Note that we allow for negative values of $s^{\prime }$, which correspond to quasi-Banach spaces $L^{s}$.

The wigner property for CL-spaces and finite-dimensional polyhedral banach spaces

2021 ◽  
pp. 1-17
Author(s):  
Dongni Tan ◽  
Xujian Huang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Phase Function ◽  
Linear Isometry ◽  
Basic Properties ◽  
Finite Dimensional ◽  
Image Position

Abstract We say that a map $f$ from a Banach space $X$ to another Banach space $Y$ is a phase-isometry if the equality \[ \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \] holds for all $x,\,y\in X$ . A Banach space $X$ is said to have the Wigner property if for any Banach space $Y$ and every surjective phase-isometry $f : X\rightarrow Y$ , there exists a phase function $\varepsilon : X \rightarrow \{-1,\,1\}$ such that $\varepsilon \cdot f$ is a linear isometry. We present some basic properties of phase-isometries between two real Banach spaces. These enable us to show that all finite-dimensional polyhedral Banach spaces and CL-spaces possess the Wigner property.

scholarly journals On Riesz operators

1969 ◽  
Vol 16 (3) ◽  
pp. 227-232 ◽  
Author(s):  
J. C. Alexander
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Operators ◽  
Image Position ◽  
Schauder’S Theorem

In (4) Vala proves a generalization of Schauder's theorem (3) on the compactness of the adjoint of a compact linear operator. The particular case of Vala's result that we shall be concerned with is as follows. Let t1 and t2 be non-zero bounded linear operators on the Banach spaces Y and X respectively, and denote by 1T2 the operator on B(X, Y) defined by

scholarly journals Fourier multipliers on spaces of distributions

1986 ◽  
Vol 29 (3) ◽  
pp. 309-327 ◽  
Author(s):  
W. Lamb
Keyword(s):  
Banach Spaces ◽  
Growth Conditions ◽  
Fourier Multipliers ◽  
Bounded Operators ◽  
Vertical Strip ◽  
One Dimensional ◽  
The One ◽  
Image Position ◽  
Complex Valued

In [8], Rooney defines a class of complex-valued functions ζ each of which is analytic in a vertical strip α(ζ)< Res < β(ζ) in the complex s-plane and satisfies certain growth conditions as |Im s| →∞ along fixed lines Re s = c lying within this strip. These conditions mean that the functionsfulfil the requirements of the one-dimensional Mihlin-Hörmander theorem (see [6, p. 417]) and so can be regarded as Fourier multipliers for the Banach spaces . Consequently, each function gives rise to a family of bounded operators W[ζ,σ] σ ∈(α(ζ),β(ζ)), on , 1<p<∞.

scholarly journals ADDITIVE FUNCTIONAL INEQUALITIES AND DERIVATIONS ON HILBERT C*-MODULES

2013 ◽  
Vol 55 (2) ◽  
pp. 341-348 ◽  
Author(s):  
FRIDOUN MORADLOU
Keyword(s):  
Banach Spaces ◽  
Functional Inequality ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Formula Group ◽  
Image Position

AbstractIn this paper we investigate the following functional inequality $ \begin{eqnarray*} \| f(x-y-z) - f(x-y+z) + f(y) +f(z)\| \leq \|f(x+y-z) - f(x)\| \end{eqnarray*}$ in Banach spaces, and employing the above inequality we prove the generalized Hyers–Ulam stability of derivations in Hilbert C*-modules.

A characterization of M-Summands

1974 ◽  
Vol 76 (1) ◽  
pp. 157-159 ◽  
Author(s):  
Richard Evans
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Structure Theory ◽  
Linear Projection ◽  
Image Position

In the structure theory of Banach spaces as developed in (1), an important role is played by subspaces which are the ranges of projections having norm properties akin to those of the classical Banach spaces. A linear projection e on a Banach space V is called an M-projection ifand an L-projection if, insteadA closed subspace J of V is called an M-Summand if it is the range of an M-projection and an M-Ideal if J0 is the range of an L-projection in V′. Every M-Summand is an M-Ideal but the reverse is false.

scholarly journals Simple eigenvalues and bifurcation for a multiparameter problem

1988 ◽  
Vol 31 (1) ◽  
pp. 77-88 ◽  
Author(s):  
D. F. McGhee ◽  
M. H. Sallam
Keyword(s):  
Banach Spaces ◽  
Simple Eigenvalue ◽  
Linear Mapping ◽  
Spectral Parameters ◽  
Non Linear ◽  
Multiparameter Problem ◽  
Image Position ◽  
Bifurcation Of Solutions

We are concerned with the problem of bifurcation of solutions of a non-linear multiparameter problem at a simple eigenvalue of the linearised problem.Let X and Y be real Banach spaces, and let A, Bi, i = 1, …, n∈B(X, Y). Let : Rn × X → Y be a non-linear mapping. We consider the equationwhereand λ=(λ1, λ2,…,λn) ∈ Rn is an n-tuple of spectral parameters.

scholarly journals Completely bounded isomorphisms of injective von Neumann algebras

1989 ◽  
Vol 32 (2) ◽  
pp. 317-327 ◽  
Author(s):  
Erik Christensen ◽  
Allan M. Sinclair
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Von Neumann Algebras ◽  
Linear Structure ◽  
Type I ◽  
Hausdorff Spaces ◽  
Bounded Degree ◽  
Von Neumann ◽  
Separable Predual ◽  
Image Position

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

On the Cones Associated with Biorthogonal Systems and Bases in Banach Spaces

1969 ◽  
Vol 21 ◽  
pp. 1206-1217 ◽  
Author(s):  
C. W. Mcarthur ◽  
Ivan Singer ◽  
Mark Levin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Biorthogonal System ◽  
Biorthogonal Systems ◽  
Image Position

1. Let E be a Banach space (by this we shall mean, for simplicity, a real Banach space) and (xn,fn) ({xn} ⊂ E, {fn} ⊂ E*) a biorthogonal system, such that {fn} is total on E (i.e. the relations x ∈ E,fn(x) = 0, n = 1, 2, …, imply x = 0). Then it is natural to consider the cone1which we shall call “the cone associated with the biorthogonal system (xn,fn)”. In particular, if {xn} is a basis of E and {fn} the sequence of coefficient functional associated with the basis {xn}, this cone is nothing else but2and we shall call it “the cone associated with the basis {xn}”.

Ergodic theorems for semifinite von Neumann algebras: II

1980 ◽  
Vol 88 (1) ◽  
pp. 135-147 ◽  
Author(s):  
F. J. Yeadon
Keyword(s):  
Banach Spaces ◽  
Ergodic Theorem ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Norm ◽  
Ergodic Theorems ◽  
Unbounded Operators ◽  
Von Neumann ◽  
The Mean ◽  
Image Position

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖∞ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averagesconverge in norm.

scholarly journals Duals of Banach spaces of entire functions

1990 ◽  
Vol 32 (2) ◽  
pp. 215-220 ◽  
Author(s):  
J. M. Anderson ◽  
J. Duncan
Keyword(s):  
Banach Spaces ◽  
Entire Functions ◽  
Positive Function ◽  
Image Position ◽  
Strictly Positive Function

Let w be a strictly positive function on ℂ and let , respectively denote the Banach spaces of those entire functions φ(z) with ∣φ(z)∣= O(w(z)) and ∣φ(z)∣ = o(w(z)). In this generality, these spaces may contain only constants, but for many functions w(z) these will be interesting Banach spaces with norm.We study two specific problems.

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