Lebesgue Decomposition Research Articles

On the Lebesgue decomposition of positive linear functionals

In this paper we will prove that the Lebesgue decomposition of positive linear functionals on Banach ∗ ^* -algebras corresponds with the Lebesgue decomposition of the nonnegative forms derived from the functionals.

Lebesgue Decomposition and its Uniqueness of a Signed Lattice Measure

Lebesgue Decomposition and its Uniqueness of a Signed Lattice Measure

Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $\beta^{(l)}$, each of which has rotated bounded variation, i.e., $\sum_{n=0}^\infty | e^{i\phi_l} \beta_{n+1}^{(l)} - \beta_n^{(l)} |$ is finite for some $\phi_l$. This includes discrete Schr\"odinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann type potentials. For the real line, we prove that in the Lebesgue decomposition $d\mu=f dm + d\mu_s$ of such measures, the intersection of (-2,2) with the support of $d\mu_s$ is contained in an explicit finite set S (thus, $d\mu$ has no singular continuous part), and f is continuous and non-vanishing on $(-2,2) \setminus S$. The results for the unit circle are analogous, with (-2,2) replaced by the unit circle.

Weyl-Titchmarsh Theory for Schrodinger Operators with Strongly Singular Potentials

We develop Weyl-Titchmarsh theory for Schroedinger operators with strongly singular potentials such as perturbed spherical Schroedinger operators (also known as Bessel operators). It is known that in such situations one can still define a corresponding singular Weyl m-function and it was recently shown that there is also an associated spectral transformation. Here we will give a general criterion when the singular Weyl function can be analytically extended to the upper half plane. We will derive an integral representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna function. Moreover, we will show how essential supports for the Lebesgue decomposition of the spectral measure can be obtained from the boundary behavior of the singular Weyl function. Finally, we will prove a local Borg-Marchenko type uniqueness result. Our criteria will in particular cover the aforementioned case of perturbed spherical Schroedinger operators.

Lebesgue type decompositions for nonnegative forms

A nonnegative form t on a complex linear space is decomposed with respect to another nonnegative form w : it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by t which is almost dominated by w . The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions.

Absolute continuity for operator valued completely positive maps on C∗-algebras

Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C∗-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34–54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon–Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon–Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi’s matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.

Maps on positive operators preserving Lebesgue decompositions

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map ∅: B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C + D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C) + φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every such transformation φ is of the form φ(A) = SAS∗ (A ∈ B(H)+) for some invertible bounded linear or conjugate-linear operator S on H.

The Lebesgue decomposition of the free additive convolution of two probability distributions

We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on $${\mathbb{R}}$$ only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of the free additive convolution of Borel probability measures on the real line have been described by Bercovici and Voiculescu in a previous paper.

A canonical decomposition for linear operators and linear relations

An arbitrary linear relation (multivalued operator) acting from one Hilbert space to another Hilbert space is shown to be the sum of a closable operator and a singular relation whose closure is the Cartesian product of closed subspaces. This decomposition can be seen as an analog of the Lebesgue decomposition of a measure into a regular part and a singular part. The two parts of a relation are characterized metrically and in terms of Stone’s characteristic projection onto the closure of the linear relation.

The Lebesgue Decomposition Theorem for Arbitrary Contents

The decomposition theorem named after Lebesgue asserts that certain set functions have canonical representations as sums of particular set functions called the absolutely continuous and the singular ones with respect to some fixed set function. The traditional versions are for the bounded measures with respect to some fixed measure on a σ algebra, in final form due to Hahn 1921, and for the bounded contents with respect to some fixed content on an algebra, due to Bochner-Phillips 1941 and Darst 1962. Then came the version for arbitrary measures, due to R.A.Johnson 1967 and N.Y.Luther 1968. The unpleasant fact with these versions is that each one requires its particular notions of absolutely continuous and singular constituents. It seems mysterious how a common roof for all of them could look, and therefore how a universal version for arbitrary contents could be achieved - and all that while several abstract extensions of particular versions appeared in the subsequent decades, for example due to de Lucia-Morales 2003. After these decades now the present article claims to arrive at the final aim in the original context of arbitrary contents. The article will be based on the author’s new difference formation for arbitrary contents 1999. This difference formation even furnishes simple explicit formulas for the two constituents.

Piecewise Continuous Controls in Dieudonné-Rashevsky Type Problems

This paper considers multidimensional control problems governed by a first-order PDE system. It is known that, if the structure of the problem is linear-convex, then the so-called e-maximum principle, a set of necessary optimality conditions involving a perturbation parameter e > 0, holds. Assuming that the optimal controls are piecewise continuous, we are able to drop the perturbation parameter within the conditions, proving the Pontryagin maximum principle with piecewise regular multipliers (measures). The Lebesgue and Hahn decompositions of the multipliers lead to refined maximum conditions. Our proof is based on the Baire classification of the admissible controls.

New criteria to identify spectrum

In this paper we give some new criteria for identifying the components of a probability measure, in its Lebesgue decomposition. This enables us to give new criteria to identify spectral types of self-adjoint operators on Hilbert spaces, especially those of interest.

Yosida–Hewitt and Lebesgue Decompositions of States on Orthomodular Posets

Orthomodular posets are usually used as event structures of quantum mechanical systems. The states of the systems are described by probability measures (also called states) on it. It is well known that the family of all states on an orthomodular poset is a convex set, compact with respect to the product topology. This suggests using geometrical results to study its structure. In this line, we deal with the problem of the decomposition of states on orthomodular posets with respect to a given face of the state space. For particular choices of this face, we obtain, e.g., Lebesgue-type and Yosida–Hewitt decompositions as special cases. Considering, in particular, the problem of existence and uniqueness of such decompositions, we generalize to this setting numerous results obtained earlier only for orthomodular lattices and orthocomplete orthomodular posets.

Lebesgue decomposition theorem for σ-finite signed fuzzy measures

Further research on signed fuzzy measures is made. Lebesgue decomposition theorems for σ-finite signed fuzzy measures and fuzzy measures are proved under the null-null additive condition. Thus, the relative result in classical measure theory is generalized.

Recursive distribution method for probabilistic structural integrity analysis

The Recursive Distribution (RD) method represents a time-evolution law of a joint distribution function of a random vector which follows a deterministic time-evolution law under the prescription of an initial random vector. The recursive formula for the distribution function can be decomposed into the absolutely continuous part and the singular part, which is the Lebesgue decomposition of the distribution function. In this paper, a theory of the Lebesgue decomposition of the recursive formula is discussed, and a numerical algorithm of the RD method is given. The present method is applied to a probabilistic fracture mechanics analysis for piping integrity problem. In order to calculate the probability of the unstable fracture of the pipe, it is needed to evaluate the singular part of the Lebesgue decomposition. A numerical example and its comparison with the MC method are presented.

Absolute Continuity for Curvature Measures of Convex Sets I

AbstractLet us denote by Cr(K, ·), r ∈ {0,…, d ‐ 1}, the curvature measures of a convex body K in the Euclidean space ℝd with d ≥ 2. According to Lebesgue's decomposition theorem the curvature measure of order r of K, Cr(K, ·), can be written as the sum of an absolutely continuous measure, Car(K, ·), and a singular measure, Csr(K, ·), with respect to (d ‐ 1) ‐ dimensional Hausdorff measure. For example, if K is a polytope, then Car(K, ·) = 0 and if K is sufficiently smooth, then Csr(K, ·) = 0. For a general convex body K, a description of Car(K, ·) in terms of geometric quantities is known. In the present paper, we provide a corresponding explicit representation for the singular part Csr(K, .). Further, denote by Σr (K) the set of r‐singular boundary points of the convex body K· It is known that Σr(K) has σ‐finite, but possibly infinite, r ‐ dimensional Hausdorff measure. Provided that the singular part Csr(K, ·) of the curvature measure of order r vanishes, for a given convex body K, we prove that the r ‐ dimensional Hausdorff measure of Σr (K) also vanishes. Examples show that in a certain sense this result is sharp. Analogous results are established for surface area measures and singular normal vectors.

Generalized Lebesgue decomposition of continuous functions

Generalized Lebesgue decomposition of continuous functions

On the Lebesgue decomposition of the posterior distribution with respect to the prior in regular Bayesian experiments

Given a regular Bayesian experiment we can consider the Lebesgue decomposition of the posterior distributions w.r.t. the prior. Then, by using the Lebesgue decomposition of each sampling distribution w.r.t. the predictive distribution, we show that the absolutely continuous parts of the posteriors are only determined by the absolutely continuous parts of the sampling distributions, while the singular parts of the posteriors are only determined by the singular parts.

Lebesgue decomposition by σ-null-additive set functions using ideals

Using V. Ficker’s and P. Capek’s algebraic approach by ideals a Lebesgue decomposition theorem for a wide class of non-additive set functions called null-additive set functions is obtained. In special cases decompositions theorems for ⊕-decomposable measures andk-triangular set functions are obtained.

Lebesgue and Saks decompositions of σ-finite fuzzy measures

Two kinds of absolute continuity of fuzzy measures are distinguished and relations between them are established. The scope of fuzzy measures under our consideration is limited in a family of those that are σ-finite and absolute continuous with respect to a finite autocontinuous fuzzy measure. Under this restriction, Lebesgue decomposition and Saks decomposition are proved.