Infinite Dimensional Subspace Research Articles

Uncountably infinite algebraic genericity and spaceability for sequence spaces

Let X be a topological vector space of complex-valued sequences and Y be a subset of X. We provide conditions for (X∖Y)∪{0} to contain uncountably infinitely many linearly independent dense vector subspaces of X. We also provide conditions for (X∖Y)∪{0} to contain uncountably infinitely many linearly independent closed infinite-dimensional vector subspaces of X. We apply these results to a chain of spaces containing the ℓp spaces.

Uniformly convex subspaces of measures with the kantorovich norm

In this paper, we consider signed Borel measures on a compact metric space. We study the uniform convexity of the Kantorovich norm on subspaces of the whole space of signed measures. We construct an example of an infinite-dimensional subspace of measures on which the Kantorovich norm is uniformly convex. We also obtain an example of an infinite compact set $(X, \rho)$ such that all uniformly convex subspaces of the space of measures on $X$ are finite-dimensional.

A local Wheeler-DeWitt measure for the string landscape

According to the ‘Cosmological Central Dogma’, de Sitter space can be viewed as a quantum mechanical system with a finite number of degrees of freedom, set by the horizon area. We use this assumption together with the Wheeler-DeWitt (WDW) equation to approach the measure problem of eternal inflation. Thus, our goal is to find a time-independent wave function of the universe on a total Hilbert space defined as the direct sum of a variety of subspaces: A finite-dimensional subspace for each de Sitter vacuum and an infinite-dimensional subspace for each terminal Minkowski or AdS vacuum. We argue that, to be consistent with semiclassical intuition, such a solution requires the presence of sources. These are implemented as an inhomogenous term in the WDW equation, induced by the Hartle-Hawking no-boundary or the Linde/Vilenkin tunneling proposal. Taken together, these steps unambiguously lead to what we would like to think of as a ‘Local WDW measure,’ where ‘local’ refers to the fact that the dS part of the resulting wave function describes a superposition of static patches. The global 3-sphere spatial section of the entire multiverse makes no appearance.

Collective terahertz fluctuation modes in a polariton laser

A polariton Bardeen-Cooper-Schrieffer (BCS) state in a semiconductor microcavity is an example of symmetry-broken states in open systems. Fluctuations of the order parameter are an important tool to characterize such a state. With the condensate formed by composite particles, the set of zero-momentum fluctuations spans an infinite-dimensional electron-hole mode subspace. We show that collective fluctuation modes with orbital angular momentum different from that of the order parameter can be obtained with terahertz radiation, and that a physical manifestation of such modes, which are not Higgs modes, can be terahertz gain.

Cup product in bounded cohomology of negatively curved manifolds

Let M M be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential 2 2 -form ξ ∈ Ω 2 ( M ) \xi \in \Omega ^2(M) defines a bounded cocycle c ξ ∈ C b 2 ( M ) c_\xi \in C_b^2(M) by integrating ξ \xi over straightened 2 2 -simplices. In particular Barge and Ghys [Invent. Math. 92 (1988), pp. 509–526] proved that, when M M is a closed hyperbolic surface, Ω 2 ( M ) \Omega ^2(M) injects this way in H b 2 ( M ) H_b^2(M) as an infinite dimensional subspace. We show that the cup product of any class of the form [ c ξ ] [c_\xi ] , where ξ \xi is an exact differential 2-form, and any other bounded cohomology class is trivial in H b ∙ ( M ) H_b^{\bullet }(M) .

Filters on a countable vector space

We study various combinatorial properties, and the implications between them, for filters generated by infinite-dimensional subspaces of a countable vector space. These properties are analogous to selectivity for ultrafilters on the natural numbers and st

Total and non-total suborbits for hypercyclic operators

In this note, it is proved that if X is a separable infinite dimensional Fréchet space that admits a continuous norm then, given a closed infinite dimensional subspace of X, there exists a hypercyclic operator admitting a dense orbit which in turn admits a suborbit all of whose sub-suborbits are total in the prescribed subspace. This is related to a recently published result asserting that every supercyclic vector for an operator on a Hilbert space supports a non-total suborbit. Here we also extend this result to normed spaces.

Contracting differential equations in weighted Banach spaces

Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, such analysis is generally restricted to norm-contracting systems. We develop a generalization of geodesic contraction rates to Banach spaces using a smoothly-weighted semi-inner product structure on tangent spaces. We show that negative contraction rates in bijectively weighted spaces imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach spaces to establish the existence of fixed points. We show that contraction in surjectively weighted spaces verifies non-equilibrium asymptotic properties, such as convergence to finite- and infinite-dimensional subspaces, submanifolds, limit cycles, and phase-locking phenomena. We use contraction rates in weighted Sobolev spaces to establish existence and continuous data dependence in nonlinear PDEs, and pose a method for constructing weak solutions using vanishing one-sided Lipschitz approximations. We discuss applications to control and order reduction of PDEs.

Pseudo-Fubini Real-Entire Functions on the Plane

In this note, it is proved the existence of a mathfrak {c}-dimensional vector space of real-entire functions all of whose nonzero members are non-integrable in the sense of Lebesgue but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space can be chosen to be dense in the space of all real C^infty -functions on the plane endowed with the topology of uniform convergence on compacta for all derivatives of all orders. If the condition of being entire is dropped, then a closed infinite dimensional subspace satisfying the same properties can be obtained.

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory-Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.

Topological and algebraic genericity and spaceability for an extended chain of sequence spaces

Given a pair of topological vector spaces $$X,\; Y$$ where X is a proper linear subspace of Y it is examined whether $$Y\setminus X$$ is residual in Y (topological genericity), whether $$Y\setminus X$$ contains a dense linear subspace of Y except 0 (algebraic genericity) and whether $$Y\setminus X$$ contains a closed infinite dimensional subspace of Y except 0 (spaceability). In the present paper the spaces X and Y are either sequence spaces or spaces of analytic functions on the unit disc regarded as sequence spaces via the identification of a function with the sequence of its Taylor coefficients. For the spaces under consideration we give an affirmative answer to each of these questions providing general proofs which extend previous results.

Optimal bounds on the speed of subspace evolution* *This work was partially supported by the Heisenberg–Landau Program.

By a quantum speed limit one usually understands an estimate on how fast a quantum system can evolve between two distinguishable states. The most known quantum speed limit is given in the form of the celebrated Mandelstam–Tamm inequality that bounds the speed of the evolution of a state in terms of its energy dispersion. In contrast to the basic Mandelstam–Tamm inequality, we are concerned not with a single state but with a (possibly infinite-dimensional) subspace which is subject to the Schrödinger evolution. By using the concept of maximal angle between subspaces we derive optimal bounds on the speed of such a subspace evolution. These bounds may be viewed as further generalizations of the Mandelstam–Tamm inequality. Our study includes the case of unbounded Hamiltonians.

T-duality, vertical holonomy line bundles and loop Hori formulae

This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here, we prove T-duality and Hori formulae for winding [Formula: see text]-loop spaces, which are infinite dimensional subspaces of loop spaces.

Quantum Speed Limits for Time Evolution of a System Subspace

One of the fundamental physical limits on the speed of time evolution of a quantum state is known in the form of the celebrated Mandelstam-Tamm inequality. This inequality gives an answer to the question on how fast an isolated quantum system can evolve from its initial state to an orthogonal one. In its turn, the Fleming bound is an extension of the Mandelstam-Tamm inequality that gives an optimal speed bound for the evolution between non-orthogonal initial and final states. In the present work, we are concerned not with a single state but with a whole (possibly infinite-dimensional) subspace of the system states that are subject to the Schroedinger evolution. By using the concept of maximal angle between subspaces we derive an optimal estimate on the speed of such a subspace evolution that may be viewed as a natural generalization of the Fleming bound.

The Strong Property (B) for L_p Spaces

Given a purely non-atomic, finite measure space (Omega ,Sigma ,nu ), it is proved that for every closed, infinite-dimensional subspace V of L_p(nu ) (1le p<infty ) there exists a decomposition L_p(nu )=X_1oplus X_2, such that both subspaces X_1 and X_2 are isomorphic to L_p(nu ) and both Vcap X_1 and Vcap X_2 are infinite-dimensional. Some consequences concerning dense, non-closed range operators on L_1 are derived.

ALMOST DISJOINT AND MAD FAMILIES IN VECTOR SPACES AND CHOICE PRINCIPLES

AbstractIn set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

Sub-posets in ωω and the strong Pytkeev⁎ property

Tukey order is used to compare the cofinal complexity of partially order sets (posets). We prove that there is a 2c-sized collection of sub-posets in 2ω which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of ωω is a Tukey quotient of ωω, we answer two open questions published in [14].The relation between P-base and strong Pytkeev⁎ property is investigated. Let P be a poset equipped with a second-countable topology in which every convergent sequence is bounded. Then we prove that any topological space with a P-base has the strong Pytkeev⁎ property. Furthermore, we prove that every uncountably-dimensional locally convex space (lcs) with a P-base contains an infinite-dimensional metrizable compact subspace. Examples in function spaces are given.

THE HAUSDORFF MOMENT PROBLEM IN THE LIGHT OF ILL-POSEDNESS OF TYPE I

The Hausdorf moment problem (HMP) over the unit interval in an L 2 -setting is a classical example of an ill-posed inverse problem. Since various applications can be rewritten in terms of the HMP, it has gathered significant attention in the literature. From the point of view of regularization it is of special interest because of the occurrence of a non-compact forward operator with non-closed range. Consequently, HMP constitutes one of few examples of a linear ill-posed problem of type I in the sense of Nashed. In this paper we highlight this property and its consequences, for example, the existence of an infinite-dimensional subspace of stability. On the other hand, we show conditional stability estimates for the HMP in Sobolev spaces that indicate severe ill-posedness for the full recovery of a function from its moments, because H ̈older-type stability can be excluded. However, the associated recovery of the function value at the rightmost point of the unit interval is stable of H ̈older-type in an H1 -setting. We moreover discuss stability estimates for the truncated HMP, where the forward operator becomes compact. Some numerical case studies illustrate the theoretical results and complete the paper.

P-Schatten commutators of projections

Let $${\mathcal {H}}={\mathcal {H}}_+\oplus {\mathcal {H}}_-$$ be a fixed orthogonal decomposition of the complex separable Hilbert space $${\mathcal {H}}$$ in two infinite-dimensional subspaces. We study the geometry of the set $${\mathcal {P}}^p$$ of selfadjoint projections in the Banach algebra $${\mathcal {A}}^p=\{A\in {\mathcal {B}}({\mathcal {H}}): [A,E_+]\in {\mathcal {B}}_p({\mathcal {H}})\},$$ where $$E_+$$ is the projection onto $${\mathcal {H}}_+$$ and $${\mathcal {B}}_p({\mathcal {H}})$$ is the Schatten ideal of p-summable operators ( $$1\le p <\infty$$ ). The norm in $${\mathcal {A}}^p$$ is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space $${\mathcal {P}}^p$$ is shown to be a differentiable $$C^\infty$$ submanifold of $${\mathcal {A}}^p$$ , and a homogeneous space of the group of unitary operators in $${\mathcal {A}}^p$$ . The connected components of $${\mathcal {P}}^p$$ are characterized, by means of a partition of $${\mathcal {P}}^p$$ in nine classes, four discrete classes, and five essential classes: (1) the first two corresponding to finite rank or co-rank, with the connected components parametrized by these ranks; (2) the next two discrete classes carrying a Fredholm index, which parametrizes their components; (3) the remaining essential classes, which are connected.