Analysis of bounded sets in a topological tensor product

The aim of this paper is to investigate the nature of bounded sets in a topological ∈-tensor product EX_∈* F of any two locally convex topological vector spaces E and F over the same scalar field K. Next, we apply the results of this investigation to the study of each of the following:

1. Totally summable families in EX_∈*F;
2. ∈-tensor product of DF-spaces;
3. Topological nature of the dual of E X_∈*F, where E and F are strong duals of Banach spaces;
4. Properties of bounded sets in an ∈-tensor product of metrizable spaces.

Forπ-tensor product, the result corresponding to (b) is well known (see Grothendieck¹) that if E and F are DF-spaces then EX_π* F and EX_π* F are DF-spaces and that the strong topology on the topological dual (EX_π*F)′, which equals the space of continuous bilinear forms on EXF, coincides with the bibounded topology. We study each of the problems from (a) to (d) for ∈-tensor products. For terminology, notations and the well-known results in the theory of topological vector spaces and the topological tensor products we refer to [1–11]. However, for convenience in presentation of the results of our investigation we give a brief survey of notations and fundamental theorems which are needed throughout this paper.     

Scalar-type spectral operators and holomorphic semigroups

We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold.

1. A generates a uniformly bounded holomorphic semigroup {e?^zA}_Re(z)≥0.
2. If \(F_N (s) \equiv \int_{ - N}^N {\tfrac{{\sin (sr)}}{r}} e^{irA} dr\) , then {‖F_N‖} _N=1 ^∞ is uniformly bounded on [0,∞) and, for all x in X, the sequence {F_N(s)x} _N=1 ^∞ converges pointwise on [0, ∞) to a vector-valued function of bounded variation.

The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup {e^?zA}_Re(z)≥0 generated by A, as follows. $$\frac{1}{2}(E\{ s\} )x + (E[0,s)) x = \mathop {\lim }\limits_{N \to \infty } \int_{ - N}^N {\frac{{\sin (sr)}}{r}} e^{irA} x\frac{{dr}}{\pi }$$ for any x in X.     

Complex Banach spaces whose duals areL 1-spaces

We prove that for a complex Banach spaceA the following properties are equivalent:

1. A ^* is isometric to anL ₁(μ)-space;
2. every family of 4 balls inA with the weak intersection property has a non-empty intersection;
3. every family of 4 balls inA such that any 3 of them have a non-empty intersection, has a non-empty intersection.

     

Die Stoppverteilungen eines Markoff-Prozesses mit lokalendlichem Potential

Consider a Markovian standard semigroup P_t, t≥o, with potential kernel U=∫P_tdt on a locally compact space E. Let μ be a finite measure on E with locally finite potential μ^U and X_t, t≥O, the process having (P_t) as transition semigroup and μ as initial law. Then for a measure ν on E the following two statements are equivalent:

1. μ^U≥νU;
2. there exists a “randomized” stopping time T such that X_T is distributed according to ν.

     

Algorithms for the vector maximization problem

We consider a convex setB inR ⁿ described as the intersection of halfspacesa _i ^T x ≦b _i (i ∈ I) and a set of linear objective functionsf _j =c _j ^T x (j ∈ J). The index setsI andJ are allowed to be infinite in one of the algorithms. We give the definition of theefficient points ofB (also called functionally efficient or Pareto optimal points) and present the mathematical theory which is needed in the algorithms. In the last section of the paper, we present algorithms that solve the following problems:

1. To decide if a given point inB is efficient.
2. To find an efficient point inB.
3. To decide if a given efficient point is the only one that exists, and if not, find other ones.
4. The solutions of the above problems do not depend on the absolute magnitudes of thec _j. They only describe the relative importance of the different activitiesx _i. Therefore we also consider $$\begin{gathered} \max G^T x \hfill \\ x efficient \hfill \\ \end{gathered} $$ for some vectorG.

     

A Remark on non-separable solutions to the Schroeder-Bernstein Problem for Banach spaces

We show that the geometric structure of Banach spaces which are solutions to the Schroeder-Bernstein Problem is very complex. More precisely, we prove that there exists a non-separable solution E to this problem such that

1. E is isomorphic to each one of its finite codimensional subspaces.
2. E has no complemented Hereditarily Indecomposable subspace.
3. E has no complemented subspace isomorphic to its square.
4. E has no non-trivial divisor.

     

Certain classes of sets in Banach spaces and a topological characterization of operators of type RN

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent:

1. T is an operator of type RN;
2. for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying;
3. for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying.

We note that the implication I)?2), is apparently new even if the operator T is weakly compact.     

Multidimensional symmetric stable processes

This paper surveys recent remarkable progress in the study of potential theory for symmetric stable processes. It also contains new results on the two-sided estimates for Green functions, Poisson kernels and Martin kernels of discontinuous symmetric α-stable process in boundedC ^1,1 open sets. The new results give explicit information on how the comparing constants depend on parameter α and consequently recover the Green function and Poisson kernel estimates for Brownian motion by passing α ↑ 2. In addition to these new estimates, this paper surveys recent progress in the study of notions of harmonicity, integral representation of harmonic functions, boundary Harnack inequality, conditional gauge and intrinsic ultracontractivity for symmetric stable processes. Here is a table of contents.

1. Introduction
2. Green function and Poisson kernel estimates

1. Estimates on balls
2. Estimates on boundedC ^1,1 domains
3. Estimates on boundedC ^1,1 open sets

1. Harmonic functions and integral representation
2. Two notions of harmonicity
3. Martin kernel and Martin boundary
4. Integral representation and uniqueness
5. Boundary Harnack principle
6. Conditional process and its limiting behavior
7. Conditional gauge and intrinsic ultracontractivity

     

The quasi-isometry classification of rank one lattices

Let X be a symmetric space—other than the hyperbolic plane—of strictly negative sectional curvature. Let G be the isometry group of X. We show that any quasi-isometry between non-uniform lattices in G is equivalent to (the restriction of) a group element of G which commensurates one lattice to the other. This result has the following corollaries:

1. Two non-uniform lattices in G are quasi-isometric if and only if they are commensurable.
2. Let Γ be a finitely generated group which is quasi-isometric to a non-uniform lattice in G. Then Γ is a finite extension of a non-uniform lattice in G.
3. A non-uniform lattice in G is arithmetic if and only if it has infinite index in its quasi-isometry group.

     

Convergence of representation averages and of convolution powers

LetS be a locally compact (σ-compact) group or semi-group, and letT(t) be a continuous representation ofS by contractions in a Banach spaceX. For a regular probability μ onS, we study the convergence of the powers of the μ-averageUx=∫T(t)xdμ(t). Our main results for random walks on a groupG are:

1. if μ is adapted and strictly aperiodic, and generates a recurrent random walk, thenU ⁿ (U-I) converges strongly to 0. In particular, the random walk is completely mixing.
2. If μ×μ is ergodic onG×G, then for every unitary representationT(.) in a Hilbert space,U ⁿ converges strongly to the orthogonal projection on the space of common fixed points. These results are proved for semigroup representations, along with some other results (previously known only for groups) which do not assume ergodicity.
3. If μ is spread-out with supportS, then $\left\| {\mu ^{n + K} - \mu ^n } \right\| \to 0$ if and only if e $ \in \overline { \cup _{j = 0}^\infty S^{ - j} S^{j + K} } .$ .

     

Coherence classes of ideals of normal lattices with applications to C(X)

Given a topological space X, Jenkins and McKnight have shown how ideals of the ring C(X) are partitioned into equivalence classes — called coherence classes — defined by declaring ideals to be equivalent if their pure parts are identical. In this paper we consider a similar partitioning of the lattice of ideals of a normal bounded distributive lattice. We then apply results obtained herein to augment some of those of Jenkins and McKnight. In particular, for Tychonoff spaces, new results include the following:

1. all members of any coherence class have the same annihilator
2. every ideal is alone in its coherence class if and only if the space is a P-space.

     

Characterization of joint ergodicity for non-commuting transformations

The study of jointly ergodic transformations, begun in [2] and [1], is continued. The main result is that, ifT ₁,T ₂, …,T _s are arbitrary measure preserving transformations of a probability space (X, ?,μ), then , if and only if the following conditions are satisfied:

1. T ₁×T ₂×…×T _s is ergodic.
2. .

     

Order Norm Completions of Cone Metric Spaces

In this article, a completion theorem for cone metric spaces and a completion theorem for cone normed spaces are proved. The completion spaces are constructed by means of an equivalence relation defined via an ordered cone norm on the Banach space E whose cone is strongly minihedral and ordered closed. This order norm has to satisfy the generalized absolute value property. In particular, if E is a Dedekind complete Banach lattice, then, together with its absolute value norm, satisfy the desired properties.     

Operator martingale decompositions and the Radon-Nikodým property in Banach spaces

We consider submartingales and uniform amarts of maps acting between a Banach lattice and a Banach lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon-Nikodým property if and only if every uniformly norm bounded martingale defined on the Chaney-Schaefer l-tensor product , where E is a suitable Banach lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach lattices Y with the Radon-Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on . Secondly, we derive a Riesz decomposition for uniform amarts of maps acting between a Banach lattice and a Banach space. This result is used to characterize Banach spaces with the Radon-Nikodým property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1<p<∞, our results yield Lp(μ,Y)-space analogues of some of the well-known results on uniform amarts in L¹(μ,Y)-spaces.     

Some relations betweens-numbers of operators on Banach spaces

We considers-number sequences {s _n (T)} _n=1 ^∞ of linear and continuous operatorsT on Banach spaces and prove product formulas for the operator ideals ? _p,u ^(s) . Furthermore, we investigate the relationship between the eigenvalues of a Riesz operator and its Hilbert numbers.     

Regular-Norm Balls can be Closed in the Strong Operator Topology

The main results obtained are:– A Dedekind complete Banach lattice Y has a Fatou norm if and only if, for any Banach lattice X, the regular-norm unit ball Ur = {T Lr(X,Y): ||T||r 1} is closed in the strong operator topology on the space of all regular operators, Lr(X,Y).– A Dedekind complete Banach lattice Y has a norm which is both Fatou and Levi if and only if, for any Banach lattice X, the regular-norm unit ball Ur is closed in the strong operator topology on the space of all bounded operators, L(X,Y).– A Banach lattice Y has a Fatou–Levi norm if and only if for every L-space X the space L(X,Y) is a Banach lattice under the operator norm.– A Banach lattice Y is isometrically order isomorphic to C(S) with the supremum norm, for some Stonean space S, if and only if, for every Banach lattice X, L(X,Y) is a Banach lattice under the operator norm.Several examples demonstrating that the hypotheses may not be removed, as well as some applications of the results obtained to the spaces of operators are also given. For instance:– If X = Lp() and Y = Lq(), where 1 < p,q < , then Lr(X,Y) is a first category subset of L(X,Y).     

On a class of σ-space with special nets

Strong CP(HCP)-netted spaces are defined and some properties are shown. In particular, the following results are shown.

1. A submetrizable space is strong CP(HCP)-netted provided that the space admits a perfect map onto a strong CP(HCP)-netted space.
2. The image of a strong CP(HCP)-netted space under a perfect map is strong CP(HCP)-netted space.
3. A stratifiable space is strong HCP-netted if the space has a countable closed cover consisting of strong HCP-netted subspaces.

     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

Thinning of point processes,revisited

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