Positive Contractions Research Articles

Dilations of one Parameter Semigroups of Positive Contractions on Lp Spaces

AbstractIt is proved in this note, that a strongly continuous semigroup of (sub)positive contractions acting on an Lp-space, for 1 < p < ∞ p ≠ 2, can be dilated by a strongly continuous group of (sub)positive isometries in a manner analogous to the dilation M. A. Akçoglu and L. Sucheston constructed for a discrete semigroup of (sub)positive contractions. From this an improvement of a von Neumann type estimation, due to R. R.Coifman and G.Weiss, on the transfer map belonging to the semigroup is deduced.

Significance of rectal contractions during multichannel urodynamic study

The influence of rectal contractions on urination was examined using multichannel urodynamic study. We reviewed a total of 246 consecutive urodynamic studies. Each study consisted of a uroflow measurement and multi-channel urodynamic study, evaluating total vesical pressure, abdominal (rectal) pressure, subtracted detrusor pressure and perianal electromyography. Rectal contractions were defined as periodic fluctuations over 5 cmH2O in abdominal pressure detected by a rectal balloon catheter. No relationship of these contractions with cough and breathing was observed. Of the 246 patients, 17 (6.9%) had a positive study for rectal contractions. The patients, who had positive rectal contractions, averaged 70-year-old were older than negative subjects averaged 62-year-old. In multichannel urodynamics, the flow rate was significantly decreased, and electromyographic activity was increased at the moment of each rectal contractions. The rectal contractions are not artifactual and may be regarded as one of causes responsible for urinary difficulty in the elderly.

Singular compactness and the Neveu decomposition

The aim of this note is to determine the Neveu decomposition for the action of an amenable semigroup ofL1 positive contractions using a criterion forsingular compactness; that is, for the singularity of all the measures in a compact convex set of functionals inL∞.

On Ergodic Averages and Absorbing Sets for Positive Contractions in L1

Let T: L 1( X) → L 1( X) be a positive contraction, that is, a linear operator satisfying || T|| ≤ 1 and T( L + 1( X)) ⊆ L + 1( X). Let S n = Σ n−1 k=0 T k , A n = (1/ n) S n , and let F = { x: limsup n→∞ A n φ( x) = ∞}, where φ is a.e. positive. First, we prove that for every sequence (ƒ n ) in L + 1( X) such that, for every m, ( T m ƒ n − ƒ n ) + tends stochastically to 0 on F, min (ƒ n , 1/ƒ n ) tends stochastically to 0 on F. This implies, in particular, the Stochastic Ergodic Theorem. Second, we study the behavior of sequences of the form a nS k(n) φ( x) to obtain decompositions of the conservative part of X into six absorbing sets.

Subsequence ergodic theorems for $L\sp p$ contractions

In this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations, or Dunford Schwartz operators, are extended to operators which are positive contractions on L P for p fixed

Le problème de Dirichlet dans les C∗-algèbres

A C ∗-algebra A, a closed ideal I, and the generator Δ of a Markov semigroup of completely positive contractions from A into itself, are given. Under the same kind of assumptions as in the classical case, a completely positive lifting Λ from the quotient algebra A I into the algebra M( A) of multipliers of A is constructed, such that, for any α in A I , its lifting Λ(α) is harmonic on I, i.e., belongs to the kernel of the generator Δ localized on I.

Approximation theorems for positive operators on L p-spaces

Let 1 < r ⩽ p < ∞. Approximation theorems for positive contractions in L ( L p(m), L r(n) ) are presented. The characterization of norm attaining extreme positive contractions is given. Note that these extreme operators are also exposed points of the positive part of the unit ball of L ( l p, l r ).

Inverse Systems of Abstract Lebesgue Spaces

We show that inverse limits exist in the category of ${L^1}$ spaces and positive linear contractions between them. This result generalizes the wellknown classical results for inverse systems of Choquet Simplexes and of $L$-balls, but our proof is simple and more purely geometrical. The result finds physical application in the study of random fields.

On the Hopf decomposition

In the paper we extend the Hopf decomposition to positive operators in Banach lattices. Let E be a Banach lattice, let E′ be the dual of E, let T: E → E be a positive operator, and let T′ be the dual of T. We define two conservative ideals I C , B C in E, E′, respectively, and two dissipative ideals I D , B D in E, E′, respectively. We prove that B C and B D are projection bands (in E′), and that if E has order continuous norm, then I C and I D are projection bands, as well. The major result of the paper is that if E has order continuous norm, and if E′ has a T′-subinvariant weak order unit e, then the projection of e on B D is also T′-subinvariant. We conclude the paper by proving that our extension is a natural one and by applying our construction to positive contractions of order complete AM-spaces with unit.

Inverse systems of abstract Lebesgue spaces

We show that inverse limits exist in the category of L 1 {L^1} spaces and positive linear contractions between them. This result generalizes the wellknown classical results for inverse systems of Choquet Simplexes and of L L -balls, but our proof is simple and more purely geometrical. The result finds physical application in the study of random fields.

A remark on the uniform zero-two law for positive contractions

A remark on the uniform zero-two law for positive contractions

On the “zero-two” law for positive contractions

Let (X, Σ,μ) be a measure space (where μ is a positive σ-additive measure) and let Lp(X,Σ,μ), 1≦p≦ + ∞ be the usual real Banach lattices.

Uniform monotonicity of norms and the strong “zero-two” law

Recently, Rainer Wittmann proved a strong “zero-two” law for positive contractions of L p -spaces, 1 ⩽ p < + ∞. In my paper Wittmann's result is extended to the class of Banach lattices having uniformly monotone norms. The main result of the paper is valid for a larger class of Banach lattices and is stronger than Katznelson and Tzafriri's “zero-two” law.

Pointwise Convergence of Alternating Sequences

Let 1 &lt; p &lt; ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence and , where is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by which is a positive linear contraction of Lq with q = p/(p — 1).Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2) exists a.e. for all.

𝐿_{𝑝}-continuity of positive semigroups on finite von Neumann algebras

Let M M be a σ \sigma -finite, finite von Neumann algebra with a faithful normal tracial state τ \tau . Let α \alpha be a one-parameter semigroup of normal positive contractions of M M . Then it is shown that α \alpha is continuous with respect to the L p {L_p} -norm ( 1 ≤ p &gt; ∞ ) (1 \leq p &gt; \infty ) induced by τ \tau if and only if it is σ \sigma -weakly continuous.

Asymptotic periodicity of the iterates of positive contractions on Banach lattices

Asymptotic periodicity of the iterates of positive contractions on Banach lattices

The zero-two law for positive contractions is valid in all Banach lattices

It is shown that the zero-two law for positive contractions, as established recently by Y. Katznelson and L. Tzafriri (1986), holds in all Banach lattices.

Jensen’s inequality for positive contractions on operator algebras

Let r be a normal semifinite trace on a von Neumann algebra, and let f be a continuous convex function on the interval [0, oo) with f (0) = 0. For a positive element a of the algebra and a positive contraction a on the algebra, the following inequality is obtained: r(f (c((a))) Zf((Ari,r,i)), where f is a convex function on the real line, A is a selfadjoint operator, and {rqi} is an orthonormal set (see [11], for example). Slightly more general, if {Pi} is a pairwise orthogonal family of projections and we set a(A) = >nPiAPi for a selfadjoint operator A, then tr a (f (A)) > tr f (a (A)) (see [2, 6, and 9]). Going further in this direction Brown and Kosaki [3] have recently proved that r(f (v*av)) < r(v* f (a)v), where r is a faithful normal semifinite trace on a von Neumann algebra, v is a contraction, and a is a positive element of the algebra. (In fact, a may be unbounded and affiliated with the algebra.) In the present paper we obtain a similar inequality for an arbitrary positive contraction a on the algebra and show that (*) r(f (a (a))) < r(a(f (a))). Here f is supposed to be a continuous convex function with f(O) = 0. (Note that (*) holds without r if f is operator convex; cf. Theorem IV.1 in [1].) The inequality (*) supports the experience that inequalities involving operators are more simply treatable provided both sides are inside of a trace (see [3 and 8] for further evidences). The idea of our proof is to approximate the operators a and a (a) by diagonal ones by means of their spectral resolution and to apply the classical Jensen's inequality in Received by the editors November 19, 1984 and, in revised form, June 24, 1985 and December 17, 1985. 1980 Mathematics Subject Cai Secondary 26A51, 47B15.

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inLp (1≦p 0 independent off) implies already limnn→∞ ‖Tnf −Tn+1n+1f ‖pp=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.

Jensen's Inequality for Positive Contractions on Operator Algebras

Jensen's Inequality for Positive Contractions on Operator Algebras