Infinite Dimensional Subspace Research Articles

Representations of the Local Area-Preserving Lie Algebra for the Klein Bottle

We give the necessary and sufficient conditions for all weight subspaces of the highest weight irreducible module V(φ) of the Lie algebra ℬ related to the Klein-bottle to be finite dimensional. We prove the Verma ℬ-module is irreducible if and only if the corresponding irreducible highest weight ℬ-module V(φ) has at least one infinite dimensional weight subspace. We also give the classification of the Harish-Chandra ℬ-module with nontrivial center.

Polygonal equalities and virtual degeneracy in [formula omitted]-spaces

Suppose 0<p⩽2 and that (Ω,μ) is a measure space for which Lp(Ω,μ) is at least two-dimensional. The central results of this paper provide a complete description of the subsets of Lp(Ω,μ) that have strict p-negative type. In order to do this we study non-trivial p-polygonal equalities in Lp(Ω,μ). These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form∑j,i=1nαjαi‖zj−zi‖pp=0 where {z1,…,zn} is a subset of Lp(Ω,μ) and α1,…,αn are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial p-polygonal equalities in Lp(Ω,μ). The cases p<2 and p=2 are substantially different and are treated separately. The case p=1 generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin. Another reason for studying non-trivial p-polygonal equalities in Lp(Ω,μ) is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if (X,d) is a metric space that has strict q-negative type for some q⩾p, then: (1) (X,d) is not isometric to any linear subspace W of Lp(Ω,μ) that contains a pair of disjointly supported non-zero vectors, and (2) (X,d) is not isometric to any subset of Lp(Ω,μ) that has non-empty interior. Furthermore, in the case p=2, it also follows that (X,d) is not isometric to any affinely dependent subset of L2(Ω,μ). More generally, we show that if (Y,ρ) is a metric space whose generalized roundness ℘ is finite and if (X,d) is a metric space that has strict q-negative type for some q⩾℘, then (X,d) is not isometric to any metric subspace of (Y,ρ) that admits a non-trivial p1-polygonal equality for some p1∈[℘,q]. It is notable in all of these statements that the metric space (X,d) can, for instance, be any ultrametric space. As a result we obtain new insights into sophisticated embedding theorems of Lemin and Shkarin. We conclude the paper by constructing some pathological infinite-dimensional linear subspaces of ℓp that do not have strict p-negative type.

Lineability and algebrability of the set of holomorphic functions with a given domain of existence

Let E be a complex Banach space and U be an open subset of E. The space of all holomorphic functions on U will be represented by H(U). We prove the following results: Theorem 1. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is lineable. This means that E(U), together with 0, contains an infinite dimensional vector subspace. Theorem 2. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is c-lineable. This means that E(U), together with 0, contains a vector subspace of dimension c, the cardinality of the continuum. (Of course Theorem 1 follows from Theorem 2, but the proof of Theorem 1 is presented in a much simpler way.) Theorem 3. Let U be a domain of existence in a separable Banach space E. Then the set E(U) of all f ∈ H(U) whose domain of existence is U is algebrable. This means that E(U), together with 0, contains a subalgebra which is generated by an infinite algebraically independent set.

A lattice of finite-type invariants of virtual knots

We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based horizontally upon the Polyak algebra and extended vertically using Manturov's functorial map $f$. For each $n$, the $n$-th vertical line in the lattice contains an infinite dimensional subspace of Kauffman finite-type invariants of degree $n$. Moreover, the lattice contains infinitely many inequivalent extensions of the Conway polynomial to long virtual knots, all of which satisfy the same skein relation. Bounds for the rank of each group in the lattice are obtained.

Tight projections of frames on infinite dimensional Hilbert spaces

We characterize the frames on an infinite dimensional separable Hilbert space that can be projected to a tight frame for an infinite dimensional subspace. A result of Casazza and Leon states that an arbitrary frame for a 2N- or (2N-1)-dimensional Hilbert space can be projected to a tight frame for an N-dimensional subspace. Surprisingly, we demonstrate a large class of frames for infinite dimensional Hilbert spaces which cannot be projected to a tight frame for any infinite dimensional subspace.

Lineability criteria, with applications

Lineability is a property enjoyed by some subsets within a vector space X. A subset A of X is called lineable whenever A contains, except for zero, an infinite dimensional vector subspace. If, additionally, X is endowed with richer structures, then the more stringent notions of dense-lineability, maximal dense-lineability and spaceability arise naturally. In this paper, several lineability criteria are provided and applied to specific topological vector spaces, mainly function spaces. Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature. Families of strict-order integrable functions, hypercyclic vectors, non-extendable holomorphic mappings, Riemann non-Lebesgue integrable functions, sequences not satisfying the Lebesgue dominated convergence theorem, nowhere analytic functions, bounded variation functions, entire functions with fast growth and Peano curves, among others, are analyzed from the point of view of lineability.

Generic results in classes of ultradifferentiable functions

Let E be a Denjoy–Carleman class of ultradifferentiable functions of Beurling type on the real line that strictly contains another class F of Roumieu type. We show that the set S of functions in E that are nowhere in the class F is large in the topological sense (it is residual), in the measure theoretic sense (it is prevalent), and that S∪{0} contains an infinite dimensional linear subspace (it is lineable). Consequences for the Gevrey classes are given. Similar results are also obtained for classes of ultradifferentiable functions defined imposing conditions on the Fourier–Laplace transform of the function.

Further properties of [formula omitted] dimension

This article establishes more properties of the ℓp dimension introduced in a previous article. Given an amenable group Γ acting by translation on ℓp(Γ), a number satisfying dimension-like properties is associated to the (usually infinite dimensional) subspaces Y of ℓp(Γ) which are invariant under the action of Γ. This may be interpreted as a von Neumann dimension for ℓp spaces. As a consequence, for p∈[1,2], if Y is a closed non-trivial Γ-invariant subspace of ℓp(Γ) and let Yn be an increasing sequence of closed Γ-invariant subspace such that ⋃Yn¯=ℓp(Γ;V), then there exist a k such that Yk∩Y≠{0}.

Application of Vertex Algebras to the Structure Theory of Certain Representations Over the Virasoro Algebra

In this paper we discuss the structure of the tensor product V'_{\alpha,\beta}\otimes L(c,h) of irreducible module from intermediate series and irreducible highest weight module over the Virasoro algebra. We generalize Zhang's irreducibility criterion, and show that irreducibility depends on the existence of integral roots of a certain polynomial, induced by a singular vector in the Verma module V(c,h). A new type of irreducible Vir-module with infinite-dimensional weight subspaces is found. We show how the existence of intertwining operator for modules over vertex operator algebra yields reducibility of V'_{\alpha ,\beta}\otimes L(c,h) which is a completely new point of view to this problem. As an example, the complete structure of the tensor product with minimal models c=-22/5 and c=1/2 is presented.

Frequently hypercyclic subspaces

We study the existence of frequently hypercyclic subspaces for a given operator, that is, the existence of closed infinite-dimensional subspaces in which every non-zero vector is frequently hypercyclic. We attack the problem with any of the three methods that have been used for hypercyclic subspaces: a constructive approach, an approach via left-multiplication operators, and an approach via tensor products.

Fréchet spaces with no infinite-dimensional Banach quotients

We exhibit examples of Fréchet Montel spaces E which have a non-reflexive Fréchet quotient but such that every Banach quotient is finite-dimensional. The construction uses a method developed by Albanese and Moscatelli and requires new ingredients. Some of the main steps in the proof are presented in Section 2. They are of independent interest and show for example that the canonical inclusion between James spaces J p ⊂ J q , 1 < p < q < ∞ , is strictly cosingular. This result requires a careful analysis of the block basic sequences of the canonical basis of the dual J p ′ of the James space J p , and permits us to show that the Fréchet space J p + = ⋂ q > p J q has no infinite-dimensional Banach quotients. Plichko and Maslyuchenko had proved that it has no infinite-dimensional Banach subspaces.

On the Krein–Milman Property and the Bade Property

Throughout this paper a study on the Krein–Milmam Property and the Bade Property is entailed reaching the following conclusions: If a real topological vector space satisfies the Krein–Milmam Property, then it is Hausdorff; if a real topological vector space satisfies the Krein–Milmam Property and is locally convex and metrizable, then all of its closed infinite dimensional vector subspaces have uncountable dimension; if a real pseudo-normed space has the Bade Property, then it is Hausdorff as well but could allow closed infinite dimensional vector subspaces with countable dimension. On other hand, we show the existence of infinite dimensional closed subspaces of ℓ∞ with the Bade Property that are not the space of convergence associated to any series in a real topological vector space. Finally, we characterize unconditionally convergent series in real Banach spaces by means of a new concept called uniform convergence of series.

Perturbation classes for semi-Fredholm operators on subprojective and superprojective spaces

We prove that the component P'+(X;Y ) of the perturbation class for the up- per semi-Fredholm operators between Banach spaces X and Y coincide with the strictly singular operators when every closed infinite dimensional subspace of X contains an infinite dimensional complemented subspace whose complement is isomorphic to X. Similarly, we prove that the com- ponent P'i(X;Y ) of the perturbation class for the lower semi-Fredholm operators coincide with the strictly cosingular operators when every infinite codimensional subspace of Y is contained in an infinite codimensional complemented subspace isomorphic to Y. We also give examples of Banach spaces satisfying the aforementioned conditions.

On the hereditary proximity to [formula omitted

In the first part of the paper we present and discuss concepts of local and asymptotic hereditary proximity to ℓ 1 . The second part is devoted to a complete separation of the hereditary local proximity to ℓ 1 from the asymptotic one. More precisely for every countable ordinal ξ we construct a separable Hereditarily Indecomposable reflexive space X ξ such that every infinite-dimensional subspace of it has Bourgain ℓ 1 -index greater than ω ξ and the space itself has no ℓ 1 -spreading model.

Absolutely continuous restrictions of a Dirac measure and non-trivial zeros of the Riemann zeta function

It is shown that the Dirac measure δ ( f ) = f ( 1 ) defined on the Banach space C ( [ 0 , 1 ] ) of complex valued continuous functions defined on the interval [ 0 , 1 ] , has an absolutely continuous restriction to an infinite dimensional subspace R of C ( [ 0 , 1 ] ) , that is f ( 1 ) = ∫ 0 1 l ( x ) f ( x ) d x , ∀ f ∈ R . Each non-trivial zero of the Riemann zeta function determines a different Radon–Nikodym density l ∈ L 1 ( [ 0 , 1 ] ) . The Riemann Hypothesis holds if and only if none of these densities belongs to L 2 ( [ 0 , 1 ] ) or if and only if R is dense in L 2 ( [ 0 , 1 ] ) .

Sous-espaces fermés de séries universelles sur un espace de Fréchet

We improve a result of Charpentier [Studia Math. 198 (2010)]. We prove that even on Fréchet spaces with a continuous norm, the existence of only one restrictively universal series implies the existence of a closed infinite-dimensional subspace of restrict

Spaceability in Banach and quasi-Banach sequence spaces

Let X be a Banach space. We prove that, for a large class of Banach or quasi-Banach spaces E of X-valued sequences, the sets E-⋃q∈Γℓq(X), where Γ is any subset of (0,∞], and E-c0(X) contain closed infinite-dimensional subspaces of E (if non-empty, of course). This result is applied in several particular cases and it is also shown that the same technique can be used to improve a result on the existence of spaces formed by norm-attaining linear operators.

Angles between infinite dimensional subspaces with applications to the Rayleigh–Ritz and alternating projectors methods

We define angles for infinite dimensional subspaces of Hilbert spaces, inspired by the work of E.J. Hannan, 1961/1962. The angles of Dixmier and Friedrichs, and the gaps are characterized. We establish connections between the angles corresponding to orthogonal complements. The sensitivity of angles with respect to subspaces is estimated. We show that the squared cosines of the angles from one subspace to another can be interpreted as Ritz values in the Rayleigh–Ritz method. The Hausdorff distance between the Ritz values, corresponding to different trial subspaces, is shown to be bounded by a constant times the gap between the subspaces. We prove a similar eigenvalue perturbation bound that involves the gap squared. An ultimate acceleration of the classical alternating projectors method is proposed. Its convergence rate is estimated in terms of the angles. We illustrate the acceleration for a domain decomposition method with a small overlap for the 1D diffusion equation.

Failure of the FPP inside an asymptotically isometric-free copy of [formula omitted

We produce an equivalent renorming of c 0 which fails to contain an asymptotically isometric copy of c 0 , yet all of its infinite-dimensional subspaces fail to have the fixed point property for nonexpansive mappings on closed, bounded, convex subsets.

On the diameter of the Banach-Mazur set

On every subspace of l ∞(ℕ) which contains an uncountable ω-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of l ∞(ℕ) is infinite. This provides a partial answer to a question asked by Johnson and Odell.