Extension of Gaussian cylinder set measures

Let E be a one-to-one continuous map of the real and separable Hilbert space H into the real and separable Hilbert space K, with E having dense range. One considers Gaussian cylinder set measures on H defined by weak covariance operators. Such cylinder set measures may be used to induce, through E, Gaussian cylinder set measures on K. The result of this paper extends a result of Sato: it characterizes the norm of the spaces K for which the induced measure extends to a probability measure on the Borel sets of K. Such a result is of interest in the robustness study of signal detection.     

Hilbert space representations of decoherence functionals and quantum measures

We show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.     

Banach ideals of operators with applications

We present some results concerning the general theory of Banach ideals of operators and give several applications to Banach space theory. We give, in Section 3, new proofs of several recent results, as well as new operator characterizations of the $\text{L}$_p-spaces of Lindenstrauss and Pelczynski. In Section 4 we prove that the space of absolutely summing operators from E to F is reflexive if both E and F are reflexive and E has the approximation property. Section 5 concerns Hilbert spaces. In particular, we compute the relative projection constant of Hilbert spaces in L_p(μ)-spaces.     

Closure in a Hilbert space of a prehilbert space Chebyshev set

Let E denote the real inner product space that is the union of all finite dimensional Euclidean spaces. There is a bounded nonconvex set S, that is a subset of E, such that each point of E has a unique nearest point in S. Let H denote the separable Hilbert space that is the completion of space E. A condition is given in order that a point in H have a unique nearest point in the closure of S. We shall also provide an example where the condition fails.     

Spectral reciprocity and matrix representations of unbounded operators

We study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on ?²(X), and the energy space HE. In particular, we prove that these operators are always essentially self-adjoint on ?²(X), but may fail to be essentially self-adjoint on HE. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the HE operators with the use of a new approximation scheme.     

Universal commutative operator algebras and transfer function realizations of polynomials

To each finite-dimensional operator space E is associated a commutative operator algebra UC(E), so that E embeds completely isometrically in UC(E) and any completely contractive map from E to bounded operators on Hilbert space extends uniquely to a completely contractive homomorphism out of UC(E). The unit ball of UC(E) is characterized by a Nevanlinna factorization and transfer function realization. Examples related to multivariable von Neumann inequalities are discussed.     

On additive and multiplicative Hilbert cubes

Given subset E of natural numbers FS(E) is defined as the collection of all sums of elements of finite subsets of E and any translation of FS(E) is said to be Hilbert cube. We can define the multiplicative analog of Hilbert cube as well. E.G. Strauss proved that for every ε>0 there exists a sequence with density >1−ε which does not contain an infinite Hilbert cube. On the other hand, Nathanson showed that any set of density 1 contains an infinite Hilbert cube. In the present note we estimate the density of Hilbert cubes which can be found avoiding sufficiently sparse (in particular, zero density) sequences. As a consequence we derive a result in which we ensure a dense additive Hilbert cube which avoids a multiplicative one.     

Some properties of frames of subspaces obtained by operator theory methods

We study the relationship among operators, orthonormal basis of subspaces and frames of subspaces (also called fusion frames) for a separable Hilbert space H. We get sufficient conditions on an orthonormal basis of subspaces E={Ei}_i∈I of a Hilbert space K and a surjective T∈L(K,H) in order that {T(Ei)}_i∈I is a frame of subspaces with respect to a computable sequence of weights. We also obtain generalizations of results in [J.A. Antezana, G. Corach, M. Ruiz, D. Stojanoff, Oblique projections and frames, Proc. Amer. Math. Soc. 134 (2006) 1031-1037], which relate frames of subspaces (including the computation of their weights) and oblique projections. The notion of refinement of a fusion frame is defined and used to obtain results about the excess of such frames. We study the set of admissible weights for a generating sequence of subspaces. Several examples are given.     

Hybrid Monte Carlo on Hilbert spaces

The Hybrid Monte Carlo (HMC) algorithm provides a framework for sampling from complex, high-dimensional target distributions. In contrast with standard Markov chain Monte Carlo (MCMC) algorithms, it generates nonlocal, nonsymmetric moves in the state space, alleviating random walk type behaviour for the simulated trajectories. However, similarly to algorithms based on random walk or Langevin proposals, the number of steps required to explore the target distribution typically grows with the dimension of the state space. We define a generalized HMC algorithm which overcomes this problem for target measures arising as finite-dimensional approximations of measures π which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert space. The key idea is to construct an MCMC method which is well defined on the Hilbert space itself.We successively address the following issues in the infinite-dimensional setting of a Hilbert space: (i) construction of a probability measure Π in an enlarged phase space having the target π as a marginal, together with a Hamiltonian flow that preserves Π; (ii) development of a suitable geometric numerical integrator for the Hamiltonian flow; and (iii) derivation of an accept/reject rule to ensure preservation of Π when using the above numerical integrator instead of the actual Hamiltonian flow. Experiments are reported that compare the new algorithm with standard HMC and with a version of the Langevin MCMC method defined on a Hilbert space.     

Hilbert,Fock and Cantorian spaces in the quantum two-slit gedanken experiment

On the one hand, a rigorous mathematical formulation of quantum mechanics requires the introduction of a Hilbert space and as we move to the second quantization, a Fock space. On the other hand, the Cantorian E-infinity approach to quantum physics was developed largely without any direct reference to the afore mentioned mathematical spaces. In the present work we utilize some novel reinterpretations of basic E^(∞) Cantorian spacetime relations in terms of the Hilbert space of quantum mechanics. Proceeding in this way, we gain a better understanding of the physico-mathematical structure of quantum spacetime which is at the heart of the paradoxical and non-intuitive outcome of the famous quantum two-slit gedanken experiment.     

Feynman formulas for nonlinear evolution equations

Transformations of measures, generalized measures, and functions generated by evolution differential equations on a Hilbert space E are studied. In particular, by using Feynman formulas, a procedure for averaging nonlinear random flows is described and an analogue of the law of large number for such flows is established (see [1, 2]).     

Unbounded graph-Laplacians in energy space, and their extensions

Our purpose is to develop computational tools for determining spectra for operators associated with infinite weighted graphs. While there is a substantial literature concerning graph-Laplacians on infinite networks, much less developed is the distinction between the operator theory for the ? ² space of the set V of vertices vs the case when the Hilbert space is defined by an energy form. A network is a triple (V,E,c) where V is a (typically countable infinite) set of vertices in a graph, with E denoting the set of edges. The function c is defined on E. It is given at the outset, symmetric and positive on E. We introduce a graph-Laplacian ??, and an energy Hilbert space $\mathcal{H}_{E}$ (both depending on c). While it is known that ?? is essentially selfadjoint on its natural domain in ? ²(V), its realization in $\mathcal{H}_{E}$ is not. We give a characterization of the Friedrichs extension of the $\mathcal{H}_{E}$ -Laplacian, and prove a formula for its computation. We obtain several corollaries regarding the diagonalization of infinite matrices. To every weighted finite-interaction countable infinite graph there is a naturally associated infinite banded matrix. With the use of the Friedrichs spectral resolution, we obtain a diagonalization formula for this family of infinite matrices. With examples we give concrete illustrations of both spectral types, and spectral multiplicities.     

Formulation of a Ritz-Galerkin type procedure for the approximate solution of the neutron transport equation

The one-group neutron transport equation is commonly given as an integrodifferential equation for the neutron density ψ(x, ω) over a domain G × S in the five-dimensional phase space $\text{E}{}_3\text{× S(¦ ω ¦ = 1)}$. In this paper we show how, by decomposing the domain of the transport operator into a complementary pair of manifolds by means of a projection operator, any transport problem can be formulated, on either manifold, in terms of a symmetric positive definite operator. We use Friedrichs' method to extend the operator to a selfadjoint operator and look for a generalized solution by minimizing a certain functional over the appropriate Hilbert space. A Ritz-Galerkin type approximation procedure is formulated, and an estimate for the difference between the exact and approximate solution is given. The procedure is illustrated for a special choice of finite dimensional subspace.     

The ∂̄ equation in DFN spaces

We obtain: “Let E be a strong dual of a complex nuclear Fréchet space (a DFN space for short) and let F be a closed C^∞ form of type (0, 1) on E. Then there exists a C^∞ function f on E as the solution of $\text{?}\text{?}\text{f=F.”}$ Since every dual nuclear complete locally convex space may be considered (from the viewpoint of its bounded sets) as an inductive limit of DFN spaces this result is immediately applicable to problems of infinite dimensional holomorphy in a setting that goes far beyond that of DFN spaces. Furthermore this result and a lemma used in its proof improve previous of C. J. Henrich and P. Raboin on the ?? equation in Hilbert or DFN spaces.     

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let M be a von Neumann algebra equipped with a normal faithful semifinite trace τ, and let E be an r.i. space on (0,∞). Let E(M) be the associated symmetric space of measurable operators. Then to any bounded linear map T from E(M) into a Hilbert space H corresponds a positive norm one functional f∈E(2)^∗(M) such that

     

A W-transform-based criterion for the existence of bounded extensions of E-operators

Let Γ(H) be the symmetric Fock space over a Hilbert space H and ε:H→Γ(H) the exponential mapping. By an E-operator we mean an operator defined on ε(H). For an E-operator A, the composition mapping Φ=A°ε is called its W-transform. In this paper, we obtain a criterion based on the W-transform for checking whether or not an E-operator becomes a bounded linear operator on the Fock space.     

When is an operator the integral of a given spectral measure?

For a closed densely defined operator T on a complex Hilbert space $\text{H}$ and a spectral measure E for $\text{H}$ of countable multiplicity q defined on a σ-algebra $\text{B}$ over an arbitrary space Λ we give three conceptually differing but equivalent answers to the question asked in the title of the paper (Theorem 1.5). We then study the simplifications which accrue when T is continuous or when q = 1 (Sect. 4). With the aid of these results we obtain necessary and sufficient conditions for T to be the integral of the spectral measure of a given group of unitary operators parametrized over a locally compact abelian group Γ (Sect. 5). Applying this result to the Hilbert space $\text{H}$ of functions which are L₂ with respect to Haar measure for Γ, we derive a generalization of Bochner's theorem on multiplication operators (Sect. 6). Some results on the multiplicity of indicator spectral measures over Γ are also obtained. When Γ = $\text{R}$ we easily deduce the classical theorem about the commutant of the associated self-adjoint operator (Sect. 7).     

Positive definite symmetric functions on linear spaces

If Φ is a positive definite function on a real linear space E of infinite dimension and Φ enjoys certain symmetry conditions we are able to show that Φ is expressible as a certain Laplace-Stieltjes transform. Conversely, if Φ is given by such a transform we can often show that Φ is positive definite on E. In particular, our results apply to the L^p spaces, 0 < p < ∞, as well as to other Orlicz spaces. We also are able to show that the only positive definite continuous sup-norm symmetric functions on C(T), the space of bounded real continuous functions on T, are constants whenever C(T) contains a sequence of functions with sup-norm one and disjoint support. Finally, we apply these ideas to obtain a result on radial exponentially convex functions on a Hilbert space.     

Interval topology on effect algebras

In this paper, we study the interval topology on effect algebras, and prove that effect algebra operation on Hilbert space effect algebra E(H) is not jointly continuous under the interval topology.     

Uniform Distribution,Discrepancy, and Reproducing Kernel Hilbert Spaces

In this paper we define a notion of uniform distribution and discrepancy of sequences in an abstract set E through reproducing kernel Hilbert spaces of functions on E. In the case of the finite-dimensional unit cube these discrepancies are very closely related to the worst case error obtained for numerical integration of functions in a reproducing kernel Hilbert space. In the compact case we show that the discrepancy tends to zero if and only if the sequence is uniformly distributed in our sense. Next we prove an existence theorem for such uniformly distributed sequences and investigate the relation to the classical notion of uniform distribution. Some examples conclude this paper.