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.

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 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.

Decay measures on locally compact abelian topological groups

2001 ◽  
Vol 131 (6) ◽  
pp. 1257-1273 ◽  
Author(s):  
I. Antoniou ◽  
S. A. Shkarin
Keyword(s):  
Banach Space ◽  
Decomposition Theorem ◽  
Topological Groups ◽  
Locally Compact ◽  
Topological Abelian Group ◽  
Absolutely Continuous ◽  
Borel Set ◽  
Direct Sum ◽  
Continuous Measures ◽  
Complex Valued

We show that the Banach space M of regular σ-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces MD and MND, where MD is the set of measures μ ∈ M whose Fourier transform vanishes at infinity and MND is the set of measures μ ∈ M such that ν ∉ MD for any ν ∈ M {0} absolutely continuous with respect to the variation |μ|. For any corresponding decomposition μ = μD + μND (μD ∈ MD and μND ∈ MND) there exist a Borel set A = A(μ) such that μD is the restriction of μ to A, therefore the measures μD and μND are singular with respect to each other. The measures μD and μND are real if μ is real and positive if μ is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from MD and MND.

scholarly journals POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

scholarly journals The Radon-Nikodym Theorem for Multimeasures

1978 ◽  
Vol 21 (2) ◽  
pp. 167-173
Author(s):  
Le van Tu
Keyword(s):  
Convex Space ◽  
Measurable Space ◽  
Locally Convex Space ◽  
The Family ◽  
Disjoint Sets ◽  
Locally Convex ◽  
Nikodym Theorem

Let (S, ℳ) be ameasurable space(that is, a setSin which is defined a σ-algebra ℳ of subsets) andXa locally convex space. A mapMfrom ℳ to the family of all non-empty subsets ofXis called a multimeasure iff for every sequence of disjoint setsAnɛ ℳ (n=1,2,… )withthe seriesconverges (in the sense of (6), p. 3) toM(A).

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.

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