scholarly journals On tensor products of nuclear operators in Banach spaces

2021 ◽  
Vol 14 (3) ◽  
pp. 187-205
Author(s):  
Oleg Reinov
Keyword(s):  
Hilbert Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Fourier Coefficients ◽  
Schatten Classes ◽  
Nuclear Operators ◽  
Vector Valued

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

Multipliers for vector valued functions

1984 ◽  
Vol 99 (1-2) ◽  
pp. 137-143
Author(s):  
José Luis Torrea
Keyword(s):  
Abelian Group ◽  
Banach Spaces ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Lebesgue Spaces ◽  
Locally Compact ◽  
Translation Invariant ◽  
Bounded Linear ◽  
Vector Valued Functions ◽  
Vector Valued

SynopsisLet G be a locally compact abelian group and let Γ be the dual of G. Let A, B be Banach spaces and Lp(G,A) the Bochner-Lebesgue spaces. We prove that the space of bounded linear translation invariant operators from L1(G, A) to LX(G, B) can be identified with the space of bounded convolution invariant (in some sense) operators and also with the space of a(A, B)-valued “weak regular” measures with the relation Tf = f *μ. (A. The existence of a function m∈ L∞ (Γ,α(A,B)), such that is also proved.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

Entropy of a Convolution Operator

2004 ◽  
Vol 11 (01) ◽  
pp. 79-85 ◽  
Author(s):  
Aleksander Urbański
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Absolutely Continuous ◽  
Doubly Stochastic ◽  
Doubly Stochastic Operator ◽  
Zero Entropy

The concept of the entropy of a doubly stochastic operator was introduced in 1999 by Ghys, Langevin, and Walczak. The idea was developed further by Kamiński and de Sam Lazaro, who also conjectured that the entropy of a convolution operator determined by a probability measure on a compact abelian group is equal to zero. We prove that this is true when the group is connected and the convolution operator is determined by a measure absolutely continuous with respect to the normalized Haar measure. Our result provides also a characterization of the set of doubly stochastic operators with non-zero entropy.

scholarly journals Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition

2020 ◽  
Vol 36 (1) ◽  
pp. 27-34 ◽  
Author(s):  
VASILE BERINDE
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Displacement Condition ◽  
Nonlinear Mappings

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

Geometric Characterizations of Hilbert Spaces

2016 ◽  
Vol 59 (4) ◽  
pp. 769-775
Author(s):  
Francisco Javier García-Pacheco ◽  
Justin R. Hill
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Geometric Properties ◽  
Generic Element

AbstractWe study some geometric properties related to the setobtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element (h, k) ∊ for H a Hilbert space.

Banach space valued weak second order stochastic processes

2021 ◽  
pp. 71-96
Author(s):  
Yûichirô Kakihara
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Abelian Group ◽  
Stochastic Processes ◽  
Compact Abelian Group ◽  
Covariance Function ◽  
Second Order ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Content Type

Banach space valued stochastic processes of weak second order on a locally compact abelian group G G is considered. These processes are recognized as operator valued processes on G G . More fully, letting U \mathfrak {U} be a Banach space and H \mathfrak {H} a Hilbert space, we study B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes. Since B ( U , H ) B(\mathfrak {U},\mathfrak {H}) has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued gramian, every B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued process has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued covariance function. Using this property we can define operator stationarity, operator harmonizability and operator V V -boundedness for B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes, in addition to scalar ones. Interrelations among these processes are obtained together with the operator stationary dilation.

scholarly journals Convex-transitive characterizations of Hilbert spaces

2009 ◽  
Vol 139 (3) ◽  
pp. 633-659 ◽  
Author(s):  
Jarno Talponen
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Weak Topology ◽  
Linear Projection ◽  
One Dimensional ◽  
Mild Conditions

We investigate real convex-transitive Banach spaces X, which admit a one-dimensional bicontractive projection P on X. Various mild conditions regarding the weak topology and the geometry of the norm are provided, which guarantee that such an X is in fact isometrically a Hilbert space. For example, if uESX is a big point such that there is a bicontractive linear projection P : X → [u] and X* is weak*-locally uniformly rotund, then X is a Hilbert space. The results obtained here are motivated by the well-known Banach—Mazur rotation problem, as well as a question posed by B. Randrianantoanina in 2002 about convex-transitive spaces.

scholarly journals Twisted Hilbert spaces

1999 ◽  
Vol 59 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Félix Cabello Sánchez
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Tongue Twister ◽  
Three Space ◽  
Space Property

A Banach space X is called a twisted sum of the Banach spaces Y and Z if it has a subspace isomorphic to Y such that the corresponding quotient is isomorphic to Z. A twisted Hilbert space is a twisted sum of Hilbert spaces. We prove the following tongue-twister: there exists a twisted sum of two subspaces of a twisted Hilbert space that is not isomorphic to a subspace of a twisted Hilbert space. In other words, being a subspace of a twisted Hilbert space is not a three-space property.

scholarly journals On vector-valued Hardy martingales and a generalized Jensen's inequality

2003 ◽  
Vol 2003 (9) ◽  
pp. 527-532
Author(s):  
Annela R. Kelly ◽  
Brian P. Kelly
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Analytic Functions ◽  
Jensen’S Inequality ◽  
Monotonicity Property ◽  
Dual Group ◽  
Analytic Vector ◽  
Jensen's Inequality ◽  
Vector Valued Functions ◽  
Vector Valued

We establish a generalized Jensen's inequality for analytic vector-valued functions on𝕋Nusing a monotonicity property of vector-valued Hardy martingales. We then discuss how this result extends to functions on a compact abelian groupG, which are analytic with respect to an order on the dual group. We also give a generalization of Helson and Lowdenslager's version of Jensen's inequality to certain operator-valued analytic functions.

Sign in / Sign up

Export Citation Format

Share Document