scholarly journals Majorization on ℓ∞ and on its closed linear subspace c, and their linear preservers

2012 ◽  
Vol 437 (9) ◽  
pp. 2340-2358 ◽  
Author(s):  
F. Bahrami ◽  
A. Bayati Eshkaftaki ◽  
S.M. Manjegani
Keyword(s):  
Linear Subspace ◽  
Closed Linear Subspace ◽  
Linear Preservers

Uniqueness of Preduals in Spaces of Operators

2014 ◽  
Vol 57 (4) ◽  
pp. 810-813 ◽  
Author(s):  
G. Godefroy
Keyword(s):  
Linear Subspace ◽  
Closed Linear Subspace ◽  
Reflexive Space ◽  
Spaces Of Operators ◽  
Weak Star

AbstractWe show that if E is a separable reflexive space, and L is a weak-star closed linear subspace of L(E) such that L ∩ K(E) is weak-star dense in L, then L has a unique isometric predual. The proof relies on basic topological arguments.

scholarly journals Biorthogonality in the real sequence spaces ℓp

1997 ◽  
Vol 40 (2) ◽  
pp. 325-330
Author(s):  
Anthony J. Felton ◽  
H. P. Rogosinski
Keyword(s):  
Linear Subspace ◽  
Sequence Spaces ◽  
Inner Product ◽  
Real Sequence ◽  
Closed Linear Subspace ◽  
The Real ◽  
Infinite Dimensional

In this paper we generalise some of the results obtained in [1] for the n-dimensional real spaces ℓp(n) to the infinite dimensional real spaces ℓp. Let p >1 with p ≠ 2, and let x be a non-zero real sequence in ℓp. Let ε(x) denote the closed linear subspace spanned by the set of all those sequences in ℓp which are biorthogonal to x with respect to the unique semi-inner-product on ℓp consistent with the norm on ℓp. In this paper we show that codim ε(x)=1 unless either x has exactly two non-zero coordinates which are equal in modulus, or x has exactly three non-zero coordinates α, β, γ with |α| ≥ |β| ≥ |γ| and |α|p > |β|p + |γ|p. In these exceptional cases codim ε(x) = 2. We show that is a linear subspace if, and only if, x has either at most two non-zero coordinates or x has exactly three non-zero coordinates which satisfy the inequalities stated above.

Spectral synthesis on zero-dimensional locally compact Abelian groups

2019 ◽  
pp. 450-456
Author(s):  
Sergey S. Platonov
Keyword(s):  
Invariant Subspace ◽  
Linear Subspace ◽  
Spectral Synthesis ◽  
Linear Span ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Closed Linear Subspace ◽  
Compact Abelian Groups ◽  
Continuous Complex ◽  
Complex Valued

Let G be a zero-dimensional locally compact Abelian group whose elements are compact, C(G) the space of continuous complex-valued functions on the group G. A closed linear subspace H⊆ C(G) is called invariant subspace, if it is invariant with respect to translations τ_y ∶ f(x) ↦ f(x + y), y ∈ G. We prove that any invariant subspace H admits spectral synthesis, which means that H coincides with the closure of the linear span of all characters of the group G contained in H.

scholarly journals Inscription on statistical convergence of order α

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2341-2347
Author(s):  
Manasi Mandal ◽  
Mandobi Banerjee
Keyword(s):  
Statistical Convergence ◽  
Linear Subspace ◽  
Closed Subspace ◽  
Acta Math ◽  
Sequence Spaces ◽  
Closed Linear Subspace ◽  
Closed Set ◽  
Remarkable Result ◽  
Convergent Sequences

In this article we recall a remarkable result stated as "For a fixed ?, 0 < ? ? 1, the set of all bounded statistically convergent sequences of order ? is a closed linear subspace of m (m is the set of all bounded real sequences endowed with the sup norm)" by Bhunia et al. (Acta Math. Hungar. 130 (1-2) (2012), 153-161) and to develop the objective of this perception we demonstrate that the set of all bounded statistically convergent sequences of order ? may not form a closed subspace in other sequence spaces. Also we determine two different sequence spaces in which the set of all statistically convergent sequences of order ? (irrespective of boundedness) forms a closed set.

scholarly journals On the problem of non-smoothness of non-reflexive second conjugate spaces

1975 ◽  
Vol 12 (3) ◽  
pp. 407-416 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Canonical Embedding ◽  
Reflexive Banach Spaces ◽  
Closed Linear Subspace ◽  
Conjugate Space ◽  
The Family

We prove that if E is a Banach space which has a subspace G such that the conjugate space G* contains a proper norm closed linear subspace V of characteristic 1, then E** is not smooth and there exist in πE(E) points of non-smoothness for E**, where πE: E → E** is the canonical embedding. We show that the spaces E having such a subspace G constitute a large proper subfamily of the family of all non-reflexive Banach spaces.

Some remarks about relaxation problems in the Calculus of Variations

1996 ◽  
Vol 126 (3) ◽  
pp. 665-675 ◽  
Author(s):  
Fabián Flores-Bazán
Keyword(s):  
Differential Operator ◽  
Weak Topology ◽  
Existence Result ◽  
Linear Subspace ◽  
Lower Semicontinuous ◽  
Variational Problems ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Closed Linear Subspace ◽  
Open Set

We study variational problems for the functional F(u) = ∫Ω f(x, u(x), Lu(x)) dx where u∈uo + V, with Vbeing any closed linear subspace of W2.P(Ω) containing W2.p.0(Ω), Ω is a bounded open set, p > 1, L is a differential operator of second order. We determine the greatest lower semicontinuous function majorised by F for the weak topology of W2.p, for its sequential version if f satisfies no coercivity assumption, showing that in both cases the relaxed functional is expressed in terms of the function ξ↦ f**(x, u, ξ). Finally, an existence result in case f (not necessarily convex) depending only on the Laplacian, is given

WHITNEY'S TYPE THEOREMS FOR INFINITE DIMENSIONAL SPACES

2000 ◽  
Vol 03 (01) ◽  
pp. 141-160
Author(s):  
S. A. SHKARIN
Keyword(s):  
Linear Extension ◽  
Linear Subspace ◽  
Extension Operator ◽  
Fréchet Space ◽  
Frechet Space ◽  
Entire Space ◽  
Closed Linear Subspace ◽  
Infinite Dimensional ◽  
Class C ◽  
Linear Extension Operator

It is proved that for any f ∈ Ck(L,ℝ), where k ∈ ℕ and L is a closed linear subspace of a nuclear Frechét space X, the function f can be extended to a function of class Ck-1 defined on the entire space X. It is also proved that for any f ∈ Ck (L, ℝ), where k ∈ℕ∪{∞} and L is a closed linear subspace of a conjugate X of a nuclear Frechét space, the function f can be extended to a function of class Ck defined on the entire space X. In addition, it is proved that under these conditions, the existence of a linear extension operator is equivalent to the complementability of the subspace.

DECOMPOSITIONS OF SPACES OF MEASURES

2008 ◽  
Vol 11 (01) ◽  
pp. 119-126
Author(s):  
IOANNIS ANTONIOU ◽  
COSTAS KARANIKAS ◽  
STANISLAV SHKARIN
Keyword(s):  
Banach Space ◽  
Absolute Continuity ◽  
Linear Subspace ◽  
Decomposition Theorem ◽  
Measurable Space ◽  
Jordan Decomposition ◽  
Closed Linear Subspace ◽  
Complex Valued ◽  
Spaces Of Measures ◽  
Nikodym Theorem

Let 𝔐 be the Banach space of σ-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of 𝔐 is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon–Nikodým theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen–Wintner purity theorem for our decompositions.

scholarly journals On Invariant Subspaces for the Shift Operator

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 743
Author(s):  
Junfeng Liu
Keyword(s):  
Hardy Space ◽  
Lebesgue Space ◽  
Linear Subspace ◽  
Shift Operator ◽  
Invariant Subspaces ◽  
Closed Linear Subspace ◽  
Large Classes ◽  
Reducing Subspaces ◽  
Hyperinvariant Subspaces ◽  
And Function

In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace M in the Hardy space H p ( D ) ( 1 ≤ p < ∞ ) is invariant under the shift operator M z on H p ( D ) if and only if it is hyperinvariant under M z , and that a closed linear subspace M in the Lebesgue space L 2 ( ∂ D ) is reducing under the shift operator M e i θ on L 2 ( ∂ D ) if and only if it is hyperinvariant under M e i θ . At the same time, we show that there are two large classes of invariant subspaces for M e i θ that are not hyperinvariant subspaces for M e i θ and are also not reducing subspaces for M e i θ . Moreover, we still show that there is a large class of hyperinvariant subspaces for M z that are not reducing subspaces for M z . Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space H 2 ( D ) , which are the analogue of the formula of the reproducing function in the Bergman space A 2 ( D ) . In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.

Compact Toeplitz operators on Bergman spaces

1998 ◽  
Vol 124 (1) ◽  
pp. 151-160 ◽  
Author(s):  
KAREL STROETHOFF
Keyword(s):  
Orthogonal Projection ◽  
Toeplitz Operators ◽  
Linear Subspace ◽  
Bergman Spaces ◽  
Berezin Transform ◽  
Open Unit ◽  
Open Unit Ball ◽  
Closed Linear Subspace ◽  
Volume Measure ◽  
Natural Description

Let Bn denote the open unit ball in Cn. We write V to denote Lebesgue volume measure on Bn normalized so that V(Bn)=1. Fix −1<γ<∞ and let Vγ denote the measure given by dVγ(z)=cγ (1−[mid ]z[mid ]2)γdV(z), for z∈Bn, where cγ=Γ(n+γ+1)/ (n!Γ(γ+1)); then Vγ(Bn)=1. The weighted Bergman space A2,γ(Bn) is the space of all analytic functions in L2(Bn, dVγ). This is a closed linear subspace of L2(Bn, dVγ). Let Pγ denote the orthogonal projection of L2(Bn, dVγ) onto A2,γ(Bn). For a function f∈L∞(Bn) the Toeplitz operator Tf is defined on A2,γ(Bn) by Tfh=Pγ(fh), for h∈A2,γ(Bn). It is clear that Tf is bounded on A2,γ(Bn) with ∥Tf∥[les ]∥f∥∞. In this paper we will consider the question for which f∈L∞(Bn) the operator Tf is compact on A2,γ(Bn). Although a complete answer has been given by the author and D. Zheng (see the next section), the condition for compactness is somewhat unnatural. In this article we will give a more natural description for compactness of Toeplitz operators with sufficiently nice symbols. We will describe compactness in terms of behaviour of the so-called Berezin transform of the symbol, which has been useful in characterizing compactness of Toeplitz operators with positive symbols (see [5, 9]). Before we can define this Berezin transform we need to introduce more notation.

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