The PT-order and the fixed point property

The PT-order, or passing through order, of a poset P is a quasi order defined on P so that ab holds if and only if every maximal chain of P which passes throug a also passes through b. We show that if P is chain complete, then it contains a subset X which has the properties that (i) each element of X is -maximal, (ii) X is a -antichain, and (iii) X is -dominating; we call such a subset a -good subset of P. A -good subset is a retract of P and any two -good subsets are order isomorphic. It is also shown that if P is chain complete, then it has the fixed point property if and only if a -good subset also has the fixed point property. Since a retract of a chain complete poset is also chain complete, the construction may be iterated transfinitely. This leads to the notion of the core of P (a -good subset of itself) which is the transfinite analogue of the core of a finite poset obtained by dismantling.Research partially supported by grants from the National Natural Science Foundation of China and The Natural Science Foundation of Shaanxi province.Research supported by NSERC grant #69-0982.     

Jordan axioms for C*-algebras

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a ²)^*=(a ^*)², aa ^* a=a³ and ab+ba2ab for alla, b inA is shown to be a C^*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C^*-algebra. Thus we obtain a purely Jordan characterization of C^*-algebras.     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

A chain complete poset with no infinite antichain has a finite core

It is well known that dismantling a finite posetP leads to a retract, called the core ofP, which has the fixed-point property if and only ifP itself has this property. The PT-order, or passing through order, of a posetP is the quasi order defined onP so thatab holds if and only if every maximal chain ofP which passes througha also passes throughb. This leads to a generalization of the dismantling procedure which works for arbitrary chain complete posets which have no infinite antichain. We prove that such a poset also has a finite core, i.e. a finite retract which reflects the fixed-point property forP.This research was written while the first author was visiting the University of Calgary.Research supported by NSERC grant #69-0982.     

Matrix functions with arbitrarily prescribed left and right partial indices

We prove that ifn2 and , are two given vectors inZ ⁿ, then there exists a matrix function inL ^n×n (T) which has a right Wiener-Hopf factorization inL ² with the partial indices and a left Wiener-Hopf factorization inL ² with the partial indices .     

A geometric construction of the K-loop of a hyperbolic space

It is well known that the homogeneous orthochronous proper Lorentzgroup is isomorphic to the proper motion group of the hyperbolic space. To each Lorentz boost \ {id} there corresponds in the hyperbolic space exactly one lineL such that fixes each of the two ends ofL . Furthermore has no fixed points but each plane containingL is fixed by . If we fix a pointo, then to each other pointa there is exactly one boosta ⁺ such thatL _a+ is the line joiningo anda anda ⁺(o)=a. The set P of points of the hyperbolic space is turned in a K-loop (P, +) bya+b:=a ⁺(b). Each line of the hyperbolic space has the representationa+Z(b) wherea, b P,b 0 andZ(b):= {x P |x+b=b+x}.Dedicated to H. Salzmann on the occasion of his 65th birthdaySupported by the NATO Scientific Affairs Division grant CRG 900103.     

Über Positive Resolventenwerte Positiver Operatoren

In this paper we study the positive resolvent values of positive operators respectively of positive elements in Banach lattice ordered algebras. In the matrix case these values are just the inverse M-matrices. One of the main results is the following: Let A be a Banach lattice ordered algebra. A positive invertible element xA is a resolvent value of a positive element yA if and only if the element x satisfies the negative principle: If aA, < 0 and xaa then xa 0.     

Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials _{[a, b} ^{]n, k} (n2, 0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers ₀, ₁, ..., _n+k, satisfying the conditions (_i–_i–1) (_{i+1{ i}< 0(i=1, ..., n +k–1), there is a unique polynomials _n,k (t) _{[a, b} ^]/n,k and pointsa=₀,<₁<...< _n+k– 1< _n+k = b (₁1 <_n, ..., _kk<_n+k–1), such that s_n,k(_i) = _i(i=0, ..., n + k), s_n,k(_i)=0 (i=1, ..., n + k–1).Translated from Matematicheskii Zametki, Vol. 11, No. 3, pp. 251–258, March, 1972.     

Ein neues Surjektivitätskriterium im Hilbertraum

LetA, B andC be linearm-accretive operators in a Hilbert space. Suppose further thatC is bounded, thatb:=inf {Re (C y, y)| y=1}>0, thatA ^–1 exists as a bounded operator and that Re (B ^* x, A ^–1 x)+a x²0 holds for allxD (B ^*) and a constanta with 0a<b. ThenCA+B is surjective, (CA+B)^–1 exists and C Ax+Bx (b–a) A x holds for allxD (A) D (B). This criterion can be applied to evolution equations of the formdu/dt+C(t)A(t)u=f(t) whereB:=d/dt.     

The Discrete Spectrum Asymptotics with Respect to a Large Coupling Constant under Strong Nonnegative Perturbations

Let A be a self-adjoint operator, let (, ) be an inner gap in the spectrum of the operator A, and let B(t) = A + tW ^* W, where the operator W(A – iI)^-1 is not necessarily bounded. Conditions are obtained under which the spectrum of B(t) in (, ) is discrete. Let N(, A, W, ), (, ), > 0, be the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to . The asymptotics of N(, A, W, ) as + is obtained in terms of the spectral asymptotics of a certain self-adjoint compact operator. Bibliography: 5 titles.