Subspaces Of Lp Research Articles

On the Finite Dimensionality of Closed Subspaces in Lp(M, dμ) ∩ Lq(M, dν)

Finding effective finite-dimensional criteria for closed subspaces in Lp, endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces, Sp⊂Lp, whose finite dimensionality is not fixed a priori and can not be checked directly. This is often the case in diverse applications, when a closed subspace Sp⊂Lp is constructed by means of some additional conditions and constraints on Lp with no direct exemplification of the functional structure of its elements. We consider a closed topological subspace, Sp(q), of the functional Banach space, Lp(M,dμ), and, moreover, one assumes that additionally, Sp(q)⊂Lq(M,dν) is subject to a probability measure ν on M. Then, we show that closed subspaces of Lp(M,dμ)∩Lq(M,dν) for q>max{1,p},p>0 are finite dimensional. The finite dimensionality result concerning the case when q>p>0 is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to Lp(M,dμ)∩Lq(M,dν).

Isomorphic classification of Lp,q-spaces, II

This is a continuation of the papers [36] and [54], in which the isomorphic classification of Lp,q, for 1<p<∞, 1≤q<∞, p≠q, on resonant measure spaces, has been obtained. The aim of this paper is to give a complete isomorphic classification of Lp,q-spaces on general σ-finite measure spaces. Towards this end, several new subspaces of Lp,q(0,1) and Lp,q(0,∞) are identified and studied.

Unconditional Schauder frames of translates in Lp(ℝd)

We show that, for 1 < p ≤ 2, the space Lp(ℝd) does not admit unconditional Schauder frames {fi, f′i}i∈ℕ where {fi} is a sequence of translates of finitely many functions and {f′i} is seminormalized. In fact, the only subspaces of Lp(ℝd) admitting such Banach frames are those isomorphic to ℓp. On the other hand, if 2 < p < +∞ and {λi}i∈ℕ ⊆ ℝd is an unbounded sequence, there is a subsequence {λmi}i∈ℕ, a function f ∈ Lp(ℝd), and a seminormalized sequence of bounded functionals {λ′i}i∈ℕ such that \({\left\{{{T_{{\lambda_{{m_i}}}}}f,f_i^\prime} \right\}_{i \in ℕ}}\) is an unconditional Schauder frame for Lp(ℝd).

On Subspaces of Cesàro Spaces

We obtain a characterization of subspaces of L p , with 1 < p < ∞, on which the L p -norm is equivalent to the norm of the Cesaro space Ces p . Also, we show that Ces p has a complemented copy of the Cesaro sequence space ces p .

The geometry of two-valued subsets of Lp -spaces

Abstract Let 𝓜(Ω, μ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ). Let B ⊂ 𝓜(Ω, μ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of L p (Ω, μ) if and only if B is an affinely independent subset of 𝓜(Ω, μ) (when 𝓜(Ω, μ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ) that have p-negative type but not q-negative type for any q &gt; p. Such sets preclude the existence of certain types of isometry into L p -spaces.

On the connected components of the conjugacy class of projectors on [formula omitted

We characterize the projectors P on a Banach space E having the property of being connected to all the other projectors obtained as a conjugation of P. Using this characterization we show an example of a Banach space where the conjugacy class of a projector splits into several path-connected components, and describe the conjugacy classes of projectors onto subspaces of ℓp⊕ℓq with p≠q.

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0&lt;p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0&lt;p0&lt;p- and Ω is an open set in Rn) and that the Fréchet envelope of Bp(·)loc(Ω) is the space B1loc(Ω). Our proofs rely heavily on the properties of the Banach envelopes of the p0-Banach local spaces of Bp(·)loc(Ω) and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for p(·)≡p, 0&lt;p&lt;1, are also given (e.g., all quasi-Banach subspace of Bploc(Ω) is isomorphic to a subspace of lp, or l∞ is not isomorphic to a complemented subspace of the Shapiro space hp-). Finally, some questions are proposed.

Cauchy integrals, Calderón projectors, and Toeplitz operators on uniformly rectifiable domains

We develop properties of Cauchy integrals associated to a general class of first-order elliptic systems of differential operators D on a bounded, uniformly rectifiable (UR) domain Ω in a Riemannian manifold M. We show that associated to such Cauchy integrals are analogues of Hardy spaces of functions on Ω annihilated by D, and we produce projections, of Calderón type, onto subspaces of Lp(∂Ω) consisting of boundary values of elements of such Hardy spaces. We consider Toeplitz operators associated to such projections and study their index properties. Of particular interest is a “cobordism argument,” which often enables one to identify the index of a Toeplitz operator on a rough UR domain with that of one on a smoothly bounded domain.

AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES

AbstractLet 1 ≤ p &lt; ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.

The classical subspaces of the projective tensor products of lpand C(α) spaces, α< ω1

We completely determine the lq and C(K) spaces which are isomorphic to a subspace of lp⊗ˆπC(α), the projective tensor product of the classical lp space, 1≤p<∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α<ω1. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from lp to l1, 1≤p<∞. The first main theorem is an extension of a result of E. Oja and states that the only lq space which is isomorphic to a subspace of lp⊗ˆπC(α) with 1≤p≤q<∞ and ω≤α<ω1 is lp. The second main theorem concerning C(K) spaces improves a result of Bessaga and Pelczynski which allows us to classify, up to isomorphism, the separable spaces (X,Y) of nuclear operators, where X and Y are direct sums of lp and C(K) spaces. More precisely, we prove the following cancellation law for separable Banach spaces. Suppose that K1 and K3 are finite or countable compact metric spaces of the same cardinality and $1 (a) (lp⊕C(K1),lq⊕C(K2)) and (lp⊕C(K3),lq⊕C(K4)) are isomorphic.

Well-posedness of equations with fractional derivative via the method of sum

We study the well-posedness of the equations with fractional derivative Dαu(t) = Au(t)+ f(t) (0 ≤ t ≤ 2π), where A is a closed operator in a Banach space X, 0 < α < 1 and Dα is the fractional derivative in the sense of Weyl. Although this problem is not always well-posed in Lp(0, 2π;X) or periodic continuous function spaces Cper([0, 2π];X), we show by using the method of sum that it is well-posed in some subspaces of Lp(0, 2π;X) or Cper([0, 2π];X).

Positive definite functions and stable random vectors

We say that a random vector X = (X1, …, Xn) in ℝn is an n-dimensional version of a random variable Y if, for any a ∈ ℝn, the random variables ΣaiXi and γ(a)Y are identically distributed, where γ: ℝn → [0,∞) is called the standard of X. An old problem is to characterize those functions γ that can appear as the standard of an n-dimensional version. In this paper, we prove the conjecture of Lisitsky that every standard must be the norm of a space that embeds in L0. This result is almost optimal, as the norm of any finite-dimensional subspace of Lp with p ∈ (0, 2] is the standard of an n-dimensional version (p-stable random vector) by the classical result of P. Levy. An equivalent formulation is that if a function of the form f(‖ · ‖K) is positive definite on ℝn, where K is an origin symmetric star body in ℝn and f: ℝ → ℝ is an even continuous function, then either the space (ℝn, ‖·‖K) embeds in L0 or f is a constant function. Combined with known facts about embedding in L0, this result leads to several generalizations of the solution of Schoenberg’s problem on positive definite functions.

FLAT SETS, ℓp-GENERATING AND FIXING c0 IN THE NONSEPARABLE SETTING

AbstractWe define asymptotically p-flat and innerly asymptotically p-flat sets in Banach spaces in terms of uniform weak* Kadec–Klee asymptotic smoothness, and use these concepts to characterize weakly compactly generated (Asplund) spaces that are c0(ω1)-generated or ℓp(ω1)-generated, where p∈(1,∞). In particular, we show that every subspace of c0(ω1) is c0(ω1)-generated and every subspace of ℓp(ω1) is ℓp(ω1)-generated for every p∈(1,∞). As a byproduct of the technology of projectional resolutions of the identity we get an alternative proof of Rosenthal’s theorem on fixing c0(ω1).

Compact operators which factor through subspaces of lp

AbstractLet 1 ≤ p ≤ ∞. A subset K of a Banach space X is said to be relatively p ‐compact if there is an 〈xn 〉 ∈ ls p (X) such that for every k ∈ K there is an 〈αn 〉 ∈ lp ′ such that k = σ∞n=1 αn xn . A linear operator T: X → Y is said to be p ‐compact if T (Ball (X)) is relatively p ‐compact in Y. The set of all p ‐compact operators Kp (X, Y) from X to Y is a Banach space with a suitable factorization norm κp and (Kp , κp ) is a Banach operator ideal. In this paper we investigate the dual operator ideal (Kd p , κd p ). It is shown that κd p (T) = πp (T) for all T ∈ B (X, Y) if either X or Y is finite‐dimensional. As a consequence it is proved that the adjoint ideal of Kd p is Ip ′, the ideal of p ′‐integral operators. Further, a composition/decomposition theorem Kd p = Πp K is proved which also yields that (Πmin p )inj = Kd p . Finally, we discuss the density of finite rank operators in Kd p and give some examples for different values of p in this respect. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Unconditional basic sequences and homogeneous Hilbertian subspaces of non-commutative $L_p$ spaces

Suppose A is a von Neumann algebra with a normal faithful normalized trace T. We prove that if E is a homogeneous Hilbertian subspace of Lp (τ) (1 ≤ p < ∞) such that the norms induced on E by Lp (τ) and L 2 (τ) are equivalent, then E is completely isomorphic to the subspace of L p ([0, 1]) spanned by Rademacher functions. Consequently, any homogeneous subspace of L p (τ) is completely isomorphic to the span of Rademacher functions in L p ([0,1]). In particular, this applies to the linear span of operators satisfying the canonical anti-commutation relations. We also show that the real interpolation space (R, C) θ,p embeds completely isomorphically into L p (R) (R is the hyperfinite III factor) for any 1 < p < 2 and θ ∈ (0,1).

Subspaces of L p , p &gt; 2, with unconditional bases have equivalent partition and weight norms

In this note we give a simple proof that every subspace of Lp, 2 < p < ∞, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently Lp has a norm determined by partitions and weights.

New models for locating a moving service facility

In this paper we analyze a new location problem which is a generalization of the well-known single facility location model. This extension consists of introducing a general objective function and replacing fixed locations by trajectories. We prove that the problem is well-stated and solvable. A Weiszfeld type algorithm is proposed to solve this generalized dynamic single facility location problem on Lp spaces of functions, with p ∈(1,2]. We prove global convergence of our algorithm once we have assumed that the set of demand functions and the initial step function belong to a subspace of Lp called Sobolev space. Finally, examples are included illustrating the application of the model to generalized regression analysis and the convergence of the proposed algorithm. The examples also show that the pointwise extension of the algorithm does not have to converge to an optimal solution of the considered problem while the proposed algorithm does.

Square functions and H∞ calculus on subspaces of Lp and on Hardy spaces

Let X be a (closed) subspace of L p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H1(ℝ N ) is the usual Hardy space, for an appropriate choice of || || F . For example if N=1, the right choice is the sum for h ∈ H1(ℝ), where H denotes the Hilbert transform.

The Long Homology Sequence for Quasi-Banach Spaces, with Applications

We establish the existence of long homology sequences in the category of quasi-Banach spaces, with values in a certain category of topological vector spaces. We derive from that new results about the structure of twisted sums of quasi-Banach and Banach spaces. Sample result: let A and B be subspaces of Lp, with 0 < p < 1. If Lp/A and Lp/B are isomorphic then so are A* and B*. In particular, if A and B are finite dimensional, then Lp/A and Lp/B are isomorphic if and only if dim(A) = dim(B).

Characterizations of Dichotomies of Evolution Equations on the Half-Line

In this paper we investigate the characterization of dichotomies of an evolution family U=(U(t,s))t≥s≥0 of bounded linear operators on Banach space X. We introduce operators I0 and IX on subspaces of Lp(R+,X) using the integral equation u(t)=U(t,s)u(s)+∫tsU(t,ξ)f(ξ)dξ. The exponential and ordinary dichotomies of U are characterized by properties of I0,IX.