Hilbert Space Open Image Research Articles

On the Asymptotic Distribution of a Spectrum of Polynomial Bundles of Differential Operators

We study the asymptotic distribution of eigenvalues of polynomial operator bundles, i.e., of operator polynomials P(z) = P0 + zP1 + · · · zmPm, where z is a spectral parameter and Pj , j = 0, 1, . . . ,m, are linear operators in the Hilbert space Open image in new window.

Lie Triple Derivations of CSL Algebras

Let Open image in new window be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space Open image in new window with Open image in new window, and Open image in new window the associated CSL algebra. It is proved that every Lie triple derivation from Open image in new window into any σ-weakly closed algebra Open image in new window containing Open image in new window is of the form X→XT−TX+h(X)I, where Open image in new window and h is a linear mapping from Open image in new window into ℂ such that h([[A,B],C])=0 for all Open image in new window.

Maximality of Positive Operator-valued Measures

A positive operator-valued measure is a (weak-star) countably additive set function from a σ-field Σ to the space of nonnegative bounded operators on a separable complex Hilbert space Open image in new window . Such functions can be written as M = V*E(·)V in which E is a spectral measure acting on a complex Hilbert space Open image in new window and V is a bounded operator from Open image in new window to Open image in new window such that the only closed linear subspace of Open image in new window , containing the range of V and reducing E (Σ), is Open image in new window itself. Attention is paid to an existing notion of maximality for positive operator-valued measures. The purpose of this paper is to show that M is maximal if and only if E, in the above representation of M, generates a maximal commutative von Neumann algebra.

Segal-Bargmann transforms associated with finite Coxeter groups

Using a polarization of a suitable restriction map, and heat-kernel analysis, we construct a generalized Segal-Bargmann transform associated with every finite Coxeter group G on ℝN. We find the integral representation of this transform, and we prove its unitarity. To define the Segal-Bargmann transform, we introduce a Hilbert space Open image in new window of holomorphic functions on Open image in new window with reproducing kernel equal to the Dunkl-kernel. The definition and properties of Open image in new window extend naturally those of the well-known classical Fock space. The generalized Segal-Bargmann transform allows to exhibit some relationships between the Dunkl theory in the Schrodinger model and in the Fock model. Further, we prove a branching decomposition of Open image in new window as a unitary Open image in new window-module and a general version of Hecke's formula for the Dunkl transform.

The Fermi Golden Rule and its Form at Thresholds in Odd Dimensions

Let H be a Schrodinger operator on a Hilbert space Open image in new window, such that zero is a nondegenerate threshold eigenvalue of H with eigenfunction Ψ0. Let W be a bounded selfadjoint operator satisfying 〈 Ψ0, WΨ0〈>0. Assume that the resolvent (H−z)−1 has an asymptotic expansion around z=0 of the form typical for Schrodinger operators on odd-dimensional spaces. Let H(ɛ) =H+ɛW for ɛ>0 and small. We show under some additional assumptions that the eigenvalue at zero becomes a resonance for H(ɛ), in the time-dependent sense introduced by A. Orth. No analytic continuation is needed. We show that the imaginary part of the resonance has a dependence on ɛ of the form ɛ2+(ν/2) with the integer ν≥−1 and odd. This shows how the Fermi Golden Rule has to be modified in the case of perturbation of a threshold eigenvalue. We give a number of explicit examples, where we compute the ``location'' of the resonance to leading order in ɛ. We also give results, in the case where the eigenvalue is embedded in the continuum, sharpening the existing ones.

Riesz frames and approximation of the frame coefficients

A frame is a fmaily {f i } i=1 ∞ of elements in a Hilbert space Open image in new window with the property that every element in Open image in new window can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coefficients. From the mathematical point of view this is gratifying, but for applications it is a problem that the calculation requires inversion of an operator on Open image in new window .