Set Of Self-adjoint Operators Research Articles

Schur complements of selfadjoint Krein space operators

Given a bounded selfadjoint operator W on a Krein space H and a closed subspace S of H, the Schur complement of W to S is defined under the hypothesis of weak complementability. A variational characterization of the Schur complement is given and the set of selfadjoint operators W admitting a Schur complement with these variational properties is shown to coincide with the set of S-weakly complementable selfadjoint operators.

Norm Retrievable Frames and Their Perturbation in Finite Dimensional Complex Hilbert Spaces

ABSTRACTIn this article, we study the property of norm retrievability of spanning vectors in a finite dimensional complex Hilbert space ℋ. Using the set of zero trace operators on ℋ and two sets of self-adjoint operators on ℋ denoted by 𝒮1,0 and 𝒮1,1, we present some equivalent conditions to the norm retrievable frames in ℋ. We will also show that the property of norm retrievability for n-dimensional complex Hilbert space ℋ with n≠2 is stable under enough small perturbation of the frame set only for phase retrievable frames.

Uncertainty relations for sets of self adjoint operators

AbstractUncertainty relations quantify the joint localisation in time respectively space domain and Fourier domain. For that reason they are an important tool in the design of wavelets. Uncertainty relations in higher dimension are most often realized as the tensor product of one dimensional uncertainty relations, whence these uncertainty relations are not invariant under rotations. The property of invariance with respect to rotation is however very desirable for image processing. We will give a new uncertainty relation for sets of self‐adjoint operators that yields the desired invariance in many examples. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Self-adjoint extensions of restrictions

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator S obtained by restricting the self-adjoint operator A : D (A) ⊆ H → H to the dense, closed with respect to the graph norm, subspace N ⊂ D (A). Neither the knowledge of S ∗ nor of the deficiency spaces of S is required. Typically A is a differential operator and N is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle p : E(h) → P(h) ,w here P(h) denotes the set of orthogonal projections in the Hilbert space hD (A)/N and p −1 (Π) is the set of self-adjoint operators in the range of Π. The set of self-adjoint operators in h, i.e. p −1 (1), parametrises the relatively prime extensions. Any (Π,Θ) ∈ E(h) determines a boundary condition in the domain of the corresponding extension AΠ,Θ and explicitly appears in the formula for the resolvent (−AΠ,Θ + z) −1 . The connection with both von Neumann's and Boundary Triples theories of self-adjoint extensions is explained. Some examples related to quantum graphs, to Schroperators with point interactions and to elliptic boundary value problems are given.

The calculus of operator functions and operator convexity

The paper is concerned with the Fréchet differentiability and operator convexity of the operator functions on sets of self-adjoint operators on finite-dimensional inner product spaces which are associated with real-valued functions of one or two variabl

The Stratonovich–Weyl correspondence for one-dimensional kinematical groups

The Stratonovich–Weyl correspondence is a restatement of the Moyal quantization where the phase space is a manifold and where a group of transformations acts on it transitively. The first and most important step is to define a mapping from the manifold into the set of self-adjoint operators on a Hilbert space, under suitable conditions. This mapping is called a Stratonovich–Weyl kernel. The construction of this mapping is discussed on coadjoint orbits of the one-dimensional Galilei, Poincaré, and Newton–Hooke groups as well as the two-dimensional Euclidean group.

Unitary equivalence of sets of self-adjoint operators

In this paper we show that the same problem contains the problem of the classification of triples {Pl, P2, P3} of orthogonal projections, where Pl ± P2- We show, moreover, that classification up to unitary equivalence of pairs of bounded self-adjoint operators would permit giving a classification up to unitary equivalence of countable sets {Ak}k~__l of bounded operators in H whose norms are jointly bounded. This confirms once again the complexity of the classification of pairs of self-adjoint operators ,$ and hence also of the corresponding triples of orthogonal projections. 1. Given any set of self-adjoint operators {Ak}k~=l whose norms are jointly bounded, we construct a pair of bounded self-adjoint operators ~lA~l, ~l.4k} in the space $~ -- ~e// as follows: 1

On statistical models for heavy nuclei

A general statistical model (L,U,P) for heavy nuclei is presented, whereL is a set of self-adjoint operators,U a group of unitary operators acting viaH → UHU−1 (H ɛL, U ɛU) as transformation group inL, andP a probability measure onL which is invariant with respect toU. Wigner’s model is discussed and criticized. A survey of all invariant probability measures for the case of locally compact and compactL is given. Two new models (a finite-dimensional and a Hilbert-space model) avoiding the disadvantages of the models treated so far are presented.

Self-adjoint abstract differential operators

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H1 whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula A≥E, B(x)≥0, and the operator B(x) is bounded for all x. An operator L0 is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) eH1 according to the formula: L0y=−y″ + Q(x)y (−∞<x<∞). It is proved that the closure of the operator L0 is a self-adjoint operator in H1 under the given assumptions.