On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

The relative rank (S : U) of a subsemigroup U of a semigroupS is the minimum size of a set V S such that U together withV generates the whole of S. As a consequence of a result ofSierpiski, it follows that for U T_X, the monoid of all self-mapsof an infinite set X, rank(T_X : U) is either 0, 1 or 2, or uncountable.In this paper, the relative ranks rank(T_X : O_X) are considered,where X is a countably infinite partially ordered set and O_Xis the endomorphism monoid of X. We show that rank(T_X : O_X) 2 if and only if either: there exists at least one elementin X which is greater than, or less than, an infinite numberof elements of X; or X has |X| connected components. Four examplesare given of posets where the minimum number of members of T_Xthat need to be adjoined to O_X to form a generating set is,respectively, 0, 1, 2 and uncountable. 2000 Mathematics SubjectClassification 08A35 (primary), 06A07, 20M20 (secondary).     

Embedding l{infty} into the Space of Bounded Operators on Certain Banach Spaces

Sufficient conditions are given on a Banach space X which ensurethat embeds in L(X), the space of all bounded linear operatorson X. A basic sequence e_n is said to be quasisubsymmetric iffor any two increasing sequences (k_n) and (l_n) of positive integerswith (k_n) (l_n) for all n, () dominates (). If a Banach space X has a seminormalized quasisubsymmetric basis then l embedsin l(X). 2000 Mathematics Subject Classification 46B28 (primary),46B03 (secondary).     

Norms on Free Topological Groups

A norm on a group G is a function N mapping G into the set ofnon-negative real numbers such that for each x and y in G, N(xy^–1) N(x)+N(y) and N(e) = 0, where e is the identity element ofG. It is shown here that if F(X) is the free topological groupon any completely regular Hausdorff space X and H is a subgroupof F(X) generated by a finite subset of X, then any norm onH can be extended to a continuous norm on F(X).     

BENFORD'S LAW FOR THE 3x + 1 FUNCTION

Benford's law (to base B) for an infinite sequence {x_k : k 1} of positive quantities x_k is the assertion that {log_B x_k: k 1} is uniformly distributed (mod 1). The 3x + 1 functionT(n) is given by T(n) = (3n + 1)/2 if n is odd, and T(n) = n/2if n is even. This paper studies the initial iterates x_k = T^(k)(x₀)for 1 k N of the 3x + 1 function, where N is fixed. It showsthat for most initial values x₀, such sequences approximatelysatisfy Benford's law, in the sense that the discrepancy ofthe finite sequence {log_B x_k : 1 k N} is small.     

The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

On a Topological Property of certain Calkin Algebras

Let X = 1^p, 1 p < , or X = c₀, B(X) be the algebra of allbounded linear operators on X, H(X) be the ideal of compactoperators in B(X), and C(X) = B(X)/H(X) be the Calkin algebraon X. For TB(X), let ||T||_c = dist(T, H(X)) be the essentialnorm of T that is the norm of T+H(X) in C(X). It is shown thatfor any operator TB(X) and any number 0 < t < 1, thereexists a closed infinite dimensional subspace Z Z X such that ||Tx|| t||T||_c, for all x Z. As a consequence, it is shown that every (not necessarily complete)submultiplicative norm on the Calkin algebra C(X) is equivalentto the quotient norm || ||_c on C(X).     

A sufficient condition for the subordination principle in ergodic optimization

Let T : X X be a continuous surjection of a topologicalspace, and let f : X be upper semi-continuous. Wewish to identify those T-invariant measures µ which maximize f dµ. We call such measures f-maximizing, and denotethe maximum by ß(f). The study of such measures andtheir properties has recently been dubbed ergodic optimization.A first step to understanding the structure of a function'smaximizing measures is to establish the following subordinationprinciple defined by T. Bousch: if µ and are T-invariantmeasures such that supp supp µ and µ is f-maximizing,then is also f-maximizing. Previous authors have approachedthis result by constructing a continuous function g : X such that f – ß(f) g T – g. We providea sufficient condition for the subordination principle whichhas advantages when the space X is noncompact.     

Polynomial symplectomorphisms

Let be the field of real or complex numbers. Let (X ²ⁿ, )be a symplectic affine space. We study the group of polynomialsymplectomorphisms of X. We show that for an arbitrary k thegroup of polynomial symplectomorphisms acts k-transitively onX. Moreover, if 2 l 2n – 2 then elements of this groupcan be characterized by polynomial automorphisms which preservethe symplectic type of all algebraic l-dimensional subvarietiesof X.     

A Graph-Theoretic Approach to the Unique Midset Property of Metric Spaces

A metric space X has the unique midset property if there isa topology-preserving metric d on X such that for every pairof distinct points x, y there is one and only one point p suchthat d(x, p) = d(y, p). The following are proved. (1) The discretespace with cardinality n has the unique midset property if andonly if n 2, 4 and n c, where c is the cardinality of thecontinuum. (2) If D is a discrete space with cardinality notgreater than c, then the countable power D^N of D has the uniquemidset property. In particular, the Cantor set and the spaceof irrational numbers have the unique midset property. A finite discrete space with n points has the unique midsetproperty if and only if there is an edge colouring of the completegraph K_n such that for every pair of distinct vertices x, ythere is one and only one vertex p such that (xp) = (yp). Letump(K_n) be the smallest number of colours necessary for sucha colouring of K_n. The following are proved. (3) For each k 0, ump(K_2k+1) = k. (4) For each k 3, k ump(K_2k) 2k–1.     

On the Compactness of the Nonlinear Evolution Operator in a Banach Space

Let X be a real Banach space and let A(t): X 2^x be dissipativefor all t(0, T). Assume that {A(t)} generates an evolution operatorU(t, s) of type (D, , f). Necessary and sufficient conditionsare given for the compactness of U(t, s) for 0 s < t <T.     

Intersections of Symbolic Powers of Prime Ideals

Let (R,m) be a local ring with prime ideals p and q such that. If R is regular and containsa field, and dim(R/p)+dim(R/q)=dim(R), then it is proved thatp^(m) q⁽ⁿ⁾ m^m+n for all positive integers m and n. This isproved using a generalization of Serre's Intersection Theoremwhich is applied to a hypersurface R/fR. The generalizationgives conditions that guarantee that Serre's bound on the intersectiondimension (R/p)+(R/q)dim(R) holds when R is nonregular.     

THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Let (n) denote the Fourier coefficients of cusp forms or thenumber of divisors of n. Estimates of the type are shown, uniformly in q X. The methods canbe extended to other arithmetic functions, for example, thenumber of representations of n as a sum of two squares or k-freenumbers. As an application, sums of the type _{n X}(n) (n) forany q-periodic function can be estimated non-trivially.     

Rangs Stables De Certaines Extensions

M. A. Rieffel [24] a introduit le rang stable topologique (tsr),pour généraliser aux C*-algèbres, le conceptde dimension de recouvrement pour les espaces compacts, affirmantainsi le principe selon lequel une C*-algèbre est ‘unespace localement compact non commutatif’. II a montréque l'on a tsr ((A)) = [dim (Â)] + 1, pour toute C*-algèbre commutative A etque trs (B/J) tsr (B), pour toute C*-algèbre B et pourtout idéal bilatère fermé J dans B (généralisantle fait que, si X est un espace compact et F un sous-ensemblefermé dans X, alors on a dim (F) dim (X), oùdim(X) est la dimension de recouvrement de X [19]). D'autrepart, le rang stable topologique peut être utilisépour obtenir des théorèmes de ‘cancellation’pour les modules projectifs, comme ceci est fait dans [25, 2].Un peu plus tard, R. H. Herman et L. N. Vaserstein [14] ontmontré que pour toute C*-algèbre unitaire A, lerang stable topologique de A et le rang stable de Basse de Acoincident, done, pour toute C*-algèbre unitaire A, onnote sr(A) cette valeur commune appelée rang stable deA. Les C*-algèbres unitaires de rang stable 1 ont étéétudiée géométriquement par M. Rørdam[27], il a montré que l'on a sr(A) = 1 si et seulementsi l'enveloppe convexe des unitaires de A est égale àla boule unité fermé de A. D'autre part, Rieffel[24] avait introduit le rang stable connexe (csr) d'une C*-algèbre,sur lequel V. Nistor [18] a publié un article trésintéressant. Mon travail dans ce papier consiste àcompléter certains travaux déjà entreprisdans les articles qui sont cités ci-dessus.     

Hypercyclic subspaces for Fréchet space operators

A continuous linear operator is hypercyclic if there is an x∈X such that the orbit {Tnx} is dense, and such a vector x is said to be hypercyclic for T. Recent progress show that it is possible to characterize Banach space operators that have a hypercyclic subspace, i.e., an infinite dimensional closed subspace H⊆X of, except for zero, hypercyclic vectors. The following is known to hold: A Banach space operator T has a hypercyclic subspace if there is a sequence (ni) and an infinite dimensional closed subspace E⊆X such that T is hereditarily hypercyclic for (ni) and Tni→0 pointwise on E. In this note we extend this result to the setting of Fréchet spaces that admit a continuous norm, and study some applications for important function spaces. As an application we also prove that any infinite dimensional separable Fréchet space with a continuous norm admits an operator with a hypercyclic subspace.     

Hochschild (Co)Homology Dimension

In 1989 Happel asked the question whether, for a finite-dimensionalalgebra A over an algebraically closed field k, gl.dim A < if and only if hch.dim A < . Here, the Hochschild cohomologydimension of A is given by hch.dim A := inf{n N₀ | dim HHⁱ(A) = 0 for i > n}. Recently Buchweitz, Green, Madsen andSolberg gave a negative answer to Happel's question. They founda family of pathological algebras A_q for which gl.dim A_q = but hch.dim A_q = 2. These algebras are pathological in manyaspects. However, their Hochschild homology behaviors are notpathological any more; indeed one has hh.dim A_q = = gl.dimA_q. Here, the Hochschild homology dimension of A is given byhh.dim A := inf{n N₀ | dim HH_i(A) = 0 for i > n}. This suggestsposing a seemingly more reasonable conjecture by replacing theHochschild cohomology dimension in Happel's question with theHochschild homology dimension: gl.dim A < if and only ifhh.dim A < if and only if hh.dim A = 0. The conjecture holdsfor commutative algebras and monomial algebras. In the casewhere A is a truncated quiver algebra, these conditions areequivalent to the condition that the quiver of A has no orientedcycles. Moreover, an algorithm for computing the Hochschildhomology of any monomial algebra is provided. Thus the cyclichomology of any monomial algebra can be read off when the underlyingfield is characteristic 0.     

Peripheral Spectra of Order-Preserving Normal Operators

Let X be a real Banach space. A set K X is called a total coneif it is closed under addition and non-negative scalar multiplication,does not contain both x and –x for any non-zero xX, andis such that K–K:= {x–y:x, yK} is dense in X. Supposethat T is a bounded linear operator on X which leaves a closedtotal cone K invariant. We denote by (T) and r(T) the spectrumand spectral radius of T. Krein and Rutman [5] showed that if T is compact, r(T) >0 and K is normal (that is, inf{||x + y||: x, y K, ||x|| =||y|| = 1} > 0), then r(T) is an eigenvalue of T with aneigenvector in K. This result was later extended by Nussbaum[6] to any bounded operator T such that r_e(T)<r(T), wherer_e(T) denotes the essential spectral radius of T, without thehypothesis of normality. The more general question of whetherr(T) (T) for all bounded operators T was answered in the negativeby Bonsall [1], who as well as giving counterexamples describeda property of K called the bounded decomposition property, whichis sufficient to guarantee that r(T) (T). More recently, Toland [8] showed that if X is a separable Hilbertspace and T is self-adjoint, then r(T) (T), without any extrahypotheses on K. In this paper we extend Toland's results tonormal operators on Hilbert spaces, removing in passing theseparability hypothesis. 1991 Mathematics Subject Classification47B65.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

A Remark on Contraction Semigroups on Banach Spaces

Let X be a complex Banach space and let J:XX^* be a duality sectionon X (that is, x,J(x)=||J(x)||||x||=||J(x)||²)=||x||²). Forany unit vector x and any (C⁰) contraction semigroup T={e^tA:t0}, Goldstein proved that if X is a Hilbert space and |T(t)x,j(x)|1 as t, then x is an eigenvector of A corresponding toa purel imaginary eigenvalue. In this article, we prove thata similar result holds if X is a strictly convex complex Banachspace.     

Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.