scholarly journals Closed subgroups generated by generic measure automorphisms

2013 ◽  
Vol 34 (3) ◽  
pp. 1011-1017 ◽  
Author(s):  
SŁAWOMIR SOLECKI
Keyword(s):  
Lebesgue Measure ◽  
Homomorphic Image ◽  
Linear Subspace ◽  
Closed Linear Subspace ◽  
Closed Group ◽  
Finite Dimensional ◽  
Measure Preserving Transformation

AbstractWe prove that for a generic measure-preserving transformation $T$, the closed group generated by $T$ is a continuous homomorphic image of a closed linear subspace of $L_0(\lambda , {\mathbb R})$, where $\lambda $ is the Lebesgue measure, and that the closed group generated by $T$contains an increasing sequence of finite-dimensional tori whose union is dense.

scholarly journals Littlewood-Paley theory on Gaussian spaces

1988 ◽  
Vol 109 ◽  
pp. 47-61 ◽  
Author(s):  
Jürgen Potthoff
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Euclidean Spaces ◽  
General Measure ◽  
Finite Dimensional

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

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.

Continuous Ergodic Extensions and Fibre Bundles

1978 ◽  
Vol 30 (02) ◽  
pp. 373-391 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Real Line ◽  
Lebesgue Space ◽  
Transformation Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Fibre Bundles ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Measure Preserving Transformation

If a locally compact group G acts as a measure preserving transformation group on a Lebesgue space X, then there is a naturally induced unitary representation of G on L2(X), and one can study the action on X by means of this representation. The situation in which the representation has discrete spectrum (i.e., is the direct sum of finite dimensional representations) and the action is ergodic was examined by von Neumann and Halmos when G is the integers or the real line [7], and by Mackey for general non-abelian G [10].

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.

Range-compatible homomorphisms over the field with two elements

2018 ◽  
Vol 34 ◽  
pp. 71-114
Author(s):  
Clément De Seguins Pazzis
Keyword(s):  
Linear Subspace ◽  
Linear Operators ◽  
Vector Spaces ◽  
Operator Spaces ◽  
Affine Subspaces ◽  
Finite Dimensional ◽  
Reflexive Operator ◽  
Recent Classification ◽  
Special Case

Let U and V be finite-dimensional vector spaces over a field K, and S be a linear subspace of the space L(U, V ) of all linear operators from U to V. A map F : S → V is called range-compatible when F(s) ∈ Im s for all s ∈ S. Previous work has classified all the range-compatible group homomorphisms provided that codimL(U,V )S ≤ 2 dim V − 3, except in the special case when K has only two elements and codimL(U,V )S = 2 dim V − 3. This article gives a thorough treatment of that special case. The results are partly based upon the recent classification of vector spaces of matrices with rank at most 2 over F2. As an application, the 2-dimensional non-reflexive operator spaces are classified over any field, and so do the affine subspaces of Mn,p(K) with lower-rank at least 2 and codimension 3.

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.

scholarly journals Spectral Enclosures and Complex Resonances for General Self-Adjoint Operators

1998 ◽  
Vol 1 ◽  
pp. 42-74 ◽  
Author(s):  
E.B. Davies
Keyword(s):  
Spectral Properties ◽  
Linear Subspace ◽  
Adjoint Operator ◽  
Dimensional Linear Subspace ◽  
Finite Dimensional ◽  
Adjoint Operators ◽  
Computation Of Eigenvalues

AbstractThis paper considers a number of related problems concerning the computation of eigenvalues and complex resonances of a general self-adjoint operator H. The feature which ties the different sections together is that one restricts oneself to spectral properties of H which can be proved by using only vectors from a pre-assigned (possibly finite-dimensional) linear subspace L.

Sufficient Fritz John optimality conditions

1975 ◽  
Vol 13 (3) ◽  
pp. 411-419 ◽  
Author(s):  
B.D. Craven
Keyword(s):  
Nonlinear Programming ◽  
Optimality Conditions ◽  
Linear Subspace ◽  
Arbitrary Dimension ◽  
Differentiable Manifold ◽  
The Other ◽  
Sufficient Optimality Conditions ◽  
Polyhedral Cones ◽  
Finite Dimensional ◽  
Fritz John Optimality Conditions

The sufficient optimality conditions, of Fritz John type, given by Gulati for finite-dimensional nonlinear programming problems involving polyhedral cones, are extended to problems with arbitrary cones and spaces of arbitrary dimension, whether real or complex. Convexity restrictions on the function giving the equality constraint can be avoided by applying a modified notion of convexity to the other functions in the problem. This approach regards the problem as optimizing on a differentiable manifold, and transforms the problem to a locally equivalent one where the optimization is on a linear subspace.

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