Compactness in translation invariant Banach spaces of distributions and compact multipliers

It is shown that for a comprehensive family of translation invariant Banach spaces (B, ∥ ∥_B) of (classes of) measurable functions or distributions on a locally compact group (including most of the spaces of interest in harmonic analysis) the following compactness criterion generalizing the well-known results due to Kolmogorov-Riesz-Weil concerning compact sets in L^p(G), 1 ? p < ∞, holds true: A closed subset M ? B is compact in B if and only if it satisfies the following conditions: (a) sup_{? ? M} ∥?∥_B < ∞; (b) $\text{? ? > 0 ?k ∈ K(G):∥k1???∥}{}_{\mathrm{B}}\text{? ?}$ for all $\text{?∈M}$; (c) $\text{?? > 0 ?h∈K(G):∥h???∥}{}_{\mathrm{B}}\text{? ?}$ for all $\text{?∈M}$. Among various applications a characterization of the space of all compact multipliers between suitable pairs of such spaces can be derived.     

Normkonvergenz von Fourierreihen in rearrangement invarianten Banachräumen

This paper studies rearrangement invariant Banach spaces of 2π-periodic functions with respect to norm convergence of Fourier series. The main result is that norm convergence takes place if and only if the space is an interpolation space of $\text{(}\text{L}{}^{\mathrm{p\prime }}\text{(T),}\text{L}{}^{\mathrm{p}}\text{(T)), 1 < p < 2,}\text{1}\text{p′}\text{+}\text{1}\text{p}\text{= 1}$, and L^p′(T) is dense in it (compare Satz 2.8). Since norm convergence and continuity of the conjugation operator are closely connected (compare Satz 2.2), this is achieved by a careful examination of this operator similar to that of D. W. Boyd for the Hilbert transform on the whole real axis. Finally, there are applications to Orlicz and Lorentz spaces.     

Invariant convex cones and causality in semisimple Lie algebras and groups

The convex cones in a simple Lie algebra $\text{G}$ invariant under the adjoint group G of $\text{G}$ are studied. Using a earlier abstract classification of such cones, we find explicit algebraic presentations of such cones in all the classical hermitian symmetric Lie algebras. (Nontrivial such cones exist only in these cases.) The G-orbits in such cones are listed. The notion of a temporal action of a Lie group with an invariant causal orientation upon a causally oriented manifold is defined. The canonical actions of such classical groups G as above on the S?hilov boundaries of the associated (tube-type) hermitian symmetric spaces are shown to be temporal actions. Corollaries are (1) the existence of nontrivial (Lie) semigroups S in the infinite-sheeted coverings $\text{G}\text{?}$ of G, which are invariant under conjugation by $\text{G}\text{?}$ and satisfy S ∩ S^?1 = {e}, and (2) the global causality (i.e. no “closed time-like curves”) of such covering groups $\text{G}\text{?}$.     

Ein operatorwertiger Hahn-Banach Satz

We generalize Arveson's extension theorem for completely positive mappings [1] to a Hahn-Banach principle for matricial sublinear functionals with values in an injective C¹-algebra or an ideal in B($\text{H}$). We characterize injective W¹-algebras by a matricial order condition. We illustrate the matricial Hahn-Banach principle by three applications: (1) Let $\text{A, B, b}$ be unital C¹-algebras, $\text{b}$ a subalgebra of $\text{A}$ and $\text{B, B}$ injective. If ?: $\text{A}$ → $\text{B}$ is a completely bounded self-adjoint $\text{b}$-bihomomorphism, then it can be expressed as the difference of two completely positive $\text{b}$-bihomomorphism. (2) Let $\text{M}$ be a W¹-algebra, containing 1_$\text{H}$, on a Hilbert space $\text{H}$. If $\text{M}$ is finite and hyperfinite, there exists an invariant expectation mapping P of B($\text{H}$) onto $\text{M}$′. P is an extension of the center trace. (3) Combes [7] proved, that a lower semicontinuous scalar weight on a C¹-algebra is the upper envelope of bounded positive functionals. We generalize this result to unbounded completely positive mappings with values in an injective W¹-algebra.     

The ∂̄ equation in DFN spaces

We obtain: “Let E be a strong dual of a complex nuclear Fréchet space (a DFN space for short) and let F be a closed C^∞ form of type (0, 1) on E. Then there exists a C^∞ function f on E as the solution of $\text{?}\text{?}\text{f=F.”}$ Since every dual nuclear complete locally convex space may be considered (from the viewpoint of its bounded sets) as an inductive limit of DFN spaces this result is immediately applicable to problems of infinite dimensional holomorphy in a setting that goes far beyond that of DFN spaces. Furthermore this result and a lemma used in its proof improve previous of C. J. Henrich and P. Raboin on the ?? equation in Hilbert or DFN spaces.     

Geometric approach to a dilation theorem

Problem: Given operators A_j ? O on Hilbert space $\text{H}$, with ΣA_j = 1, to find commuting projectors E_j on a Hilbert space $\text{H}$ ? $\text{H}$ such that (for all _j) x¹A_jy = x¹E_jy for, x, y ∈ $\text{H}$. This paper gives an explicit construction, quite different from the familiar solution.     

Projections on spaces of invariant functions

Let (X, ∑, μ) be a measure space and S be a semigroup of measure-preserving transformations T:X → X. In case μ(X) < ∞, Aribaud [1] proved the existence of a positive contractive projection P of L¹(μ) such that for every $\text{? ? L}{}^1\text{(μ), Pf}$ belongs to the closure $\text{C}{}_1\text{(?)}\text{in}\text{L}{}^1\text{(μ)}$ of the convex hull $\text{C(?)}$ of the set {$\text{? ○ T:T ? S}$}. In this paper we extend this result in three directions: we consider infinite measure spaces, vector-valued functions, and L^p spaces with 1 ? p < ∞, and prove that P is in fact the conditional expectation with respect to the σ-algebra Λ of sets of ∑ which are invariant with respect to all T?S.     

La surjectivité de l'application moyenne pour les espaces préhomogènes

Let ? be an homogeneous polynomial on $\text{R}$ⁿ. First an analog of the Borel theorem is proved for the distributions which appear at the poles of the distribution |?|^s (s ∈ $\text{C}$). If ? is the relative invariant of an irreducible regular prehomogeneous vector space, the preceding result is used to characterize the functions which are obtained by integration on the fibers of ?.     

Sampling Expansions and Interpolation in Unitarily Translation Invariant Reproducing Kernel Hilbert Spaces

Sufficient conditions are established in order that, for a fixed infinite set of sampling points on the full line, a function satisfies a sampling theorem on a suitable closed subspace of a unitarily translation invariant reproducing kernel Hilbert space. A number of examples of such reproducing kernel Hilbert spaces and the corresponding sampling expansions are given. Sampling theorems for functions on the half-line are also established in RKHS using Riesz bases in subspaces of L ²(R ⁺).     

The Banach–Saks index of rearrangement invariant spaces on [0,1]

The set of all rearrangement invariant function spaces on [0,1] having the p-Banach–Saks property has a unique maximal element for all p∈(1,2]. For p=2 this is L₂, for p∈(1,2) this is L_p,∞⁰. We compute the Banach–Saks index for the families of Lorentz spaces $\text{L}{}_{\mathrm{p,q}}\text{,}\text{1<p<∞}$, 1?q?∞, and Lorentz–Zygmund spaces L(p,α), $\text{1?p<∞,}\text{α∈}\text{R}$, extending the classical results of Banach–Saks and Kadec–Pelczynski for L_p-spaces. Our results show that the set of rearrangement invariant spaces with Banach–Saks index p∈(1,2] is not stable with respect to the real and complex interpoltaion methods. To cite this article: E.M. Semenov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Tensor products,reproducing kernels,and power series

Tensor products of holomorphic discrete series representations in reproducing kernel Hilbert spaces are decomposed by considering power series expansions of functions in the direction perpendicular to the diagonal in $\text{D}$ × $\text{D}$.     

Quantization of the action of U(k,l) on R2(K + l)

The natural action of U(k, l) on $\text{C}$^{k + l} leaves invariant a real skew non-degenerate bilinear form B, which turns $\text{C}$^{k + l} into a symplectic manifold (M, ω). The polarization F of M defined by the complex structure of $\text{C}$^{k + l} is non-positive. If L is the prequantization complex line bundle carried by (M, ω), then U(k, l) acts on the space $\text{U}$ of square-integrable $\text{L ? ΛF}{}^1$ forms on M, leaving invariant the natural non-degenerate, but non-definite, inner product ((·, ·)) on $\text{U}$. The polarization F also defines a closed, densely defined covariant differential ?? on $\text{U}$ which is U(k, l)-invariant. Let $\text{⊥}$ denote orthocomplementation with respect to ((·, ·)). It is shown that the restriction of ((·, ·)) to the U(k, l)-stable subspace ? (Ker ??) ∩ (Im ??)^$\text{⊥}$ is semi-definite and that the unitary representation of Uk, l on the Hilbert space $\text{H}$ arising from ? by dividing out null vectors is unitarily equivalent to the representation of U(k, l) obtained from the tensor product of the metap ectic and $\text{Det}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ representations of MU(k, l), the double cover of U(k, l).     

Representation and index theory for C1-algebras generated by commuting isometries

We discuss here representation and Fredholm theory for C¹-algebras generated by commuting isometries. More particularly, for n commuting isometries {V_j: 1 ? j ? n} on separable Hilbert space we give a representation resembling the well-known representation for a single isometry. Our representation permits an analysis of the C¹-algebras $\text{Ol}$=$\text{Ol}$(V_j:1?j?n) generated by the {V_j}. The commutator ideal in $\text{Ol}$ is identified precisely and, under certain additional hypotheses, the Fredholm operators in $\text{Ol}$ are also precisely determined. Finally, we obtain formulas in terms of topological data for the index of Fredholm operators in some interesting algebras of the type $\text{Ol}$(V_j:1?j?n).     

On the operation Alg Lat in finite dimensions

For any algebra $\text{U}$ of linear transformations on a finite dimensional complex vector space V, AlgLat $\text{U}$ denotes the set of transformations which leave invariant every invariant subspace of $\text{U}$. It is shown that $\text{AlgLat}\text{U}\text{=(}\text{R}\text{∩}\text{U}{}^{\mathrm{\perp }}\text{)}{}^{\mathrm{\perp }}$, where $\text{R}$ denotes the set of transformations of rank at most one and ^⊥ is an orthocomplement operation. This is used to calculate dimAlgLat$\text{U}$? dim$\text{U}$ when $\text{U}$ is the algebra generated by a nilpotent transformation.     

Limiting behavior for strongly damped nonlinear wave equations

In this paper we investigate both the existence and the limiting behavior for the equation $\text{u}{}_{\mathrm{tt}}\text{+ Au}{}_{\mathrm{t}}\text{+ Au = ?(t, u, u}{}_{\mathrm{t}}\text{)}$, where A is a sectorial operator, ? is periodic in t, and ? satisfies certain regularity and growth assumptions. In most results on limiting behavior we will assume A has compact resolvent. We consider the equation as an abstract ODE defined on a paired space X^β × X^α, 0 ? σ ? β < 1. With regard to the limiting behavior, one of our principal results will be to show that if there is a bounded set in one of the spaces considered, for which all points or trajectories enter into and remain, then there is a set J consisting of very “smooth” functions defined on all of the spaces considered, which is the maximum compact invariant set, uniformly asymptotically stable, connected, and having very strong attractivity properties in all these spaces. We will often show it attracts all points in a bounded set uniformly. We will give a few sharper results for the case where A = ?Δ. The work is motivated by recent papers of Webb and Fitzgibbon, and applies techniques found in recent papers by the author.     

Subnormal weighted translation semigroups

A weighted translation semigroup {S_t} on L²($\text{R}$₊) is defined by $\text{(S}{}_{\mathrm{t}}\text{f)(x) =}\text{(φ(x)}\text{φ(x ? t)}\text{)f(x ? t)}$ for x ? t and 0 otherwise, where φ is a continuous nonzero scalar-valued function on $\text{R}$₊. It is shown that {S_t} is subnormal if and only if φ² is the product of an exponential function and the Laplace-Stieltjes transform of an increasing function of total variation one. A necessary and sufficient condition for similarity of weighted translation semigroups is developed.     

On permutations,convex hulls,and normal operators

A spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite-dimensional space. This is used to prove the following: if A and B are normal operators on an n-dimensional complex Hilbert space H with eigenvalues given by α₁,…,α_n and β₁,…, β_n respectively, and if A ? B is also normal, then $\text{6A ? B6 ?}\text{max}\text{σ ? S}{}_{\mathrm{n}}\text{6}\text{diag}\text{(α}{}_{\mathrm{k}}\text{?β}{}_{\mathrm{\sigma (k)}}\text{)6}$ for any unitarily invariant norm on L(H).     

Essential unitarization of symplectics and applications to field quantization

Symplectic operators satisfying generic and group-invariant (spectral) positivity conditions are studied; the theory developed is applied and illustrated to determine the unique invariant frequency decomposition (equivalently, linear quantization with invariant vacuum state) of the Klein-Gordon equation in non-static spacetimes. Let (H, Ω) be any linear topological symplectic space such that there exists a real-linear and topological isomorphism of H with some complex Hilbert space carrying Ω into the imaginary part of the scalar product. Then any bounded invertible symplectic S ∈ Sp(H) (resp. bounded infinitesimally symplectic A ∈ sp(H)) which satisfies $\text{Ω(Sv, v) > 0}$ (resp. $\text{Ω(Av, v) > 0}$) for all nonzero v ω H, where S + I is invertible, is realized uniquely and constructively as a unitary (resp. skewadjoint) operator in a complex Hilbert space which depends in general on the operator and typically only densely intersects H. The essentially unique weakly and uniformly closed invariant convex cones in sp(H) are determined, extending previously known results in the finite-dimensional case. A notion of “skew-adjoint extension” of a closed semi-bounded infinitesimally symplectic operator is defined, strictly including the usual notion of positive self-adjoint extension in a complex Hilbert space; all such skew-adjoint extensions are parametrized, as in the von Neumann or Birman-Krein-Vishik theories. Finally, the unique complex Hilbertian structure—formulated on the space of solutions of the covariant Klein-Gordon equation in generic conformal perturbations of flat space—is uniquely determined by invariance under the scattering operator. The invariant Hilbert structure is explicitly calculated to first order for an infinite-dimensional class of purely time-dependent metric perturbations, and higher-order contributions are rigorously estimated.     

Sur les décompositions directes des contractions de l'espace de Hilbert

Soit H un espace de Hilbert séparable, T ∈ $\text{B}$(H) une contraction complètement nonunitaire et H₁ ? H un sous-espace fermé, invariant à T. Le but de cette Note est de trouver une condition nécessaire et suffisante pour qu'il existe un sous-espace fermé H′ ? H, invariant à T et tel que l'espace H se décompose en somme directe $\text{H = H′ ? H}{}_1$ pas nécessairement orthogonale.     

Left mean-ergodicity,fixed points,and invariant means

Recently Lau [15] generalized a result of Yeadon [25]. In the present paper we generalize Yeadon's result in another direction recasting it as a theorem of ergodic type. We call the notion of ergodicity required left mean-ergodicity and show how it relates to the mean-ergodicity of Nagel [21]. Connections with the existence of invariant means on spaces of continuous functions on semitopological semigroups S are made, connections concerning, among other things, a fixed point theorem of Mitchell [20] and Schwartz's property P of W¹-algebras [22]. For example, if $\text{M}$(S) is a certain subspace of C(S) (which was considered by Mitchell and is of almost periodic type, i.e., the right translates of a member of $\text{M}$(S) satisfy a compactness condition), then the assumption that $\text{M}$(S) has a left invariant mean is equivalent to the assumption that every representation of S of a certain kind by operators on a linear topological space X is left mean-ergodic. An analog involving the existence of a (left and right) invariant mean on $\text{M}$(S) is given, and we show our methods restrict in the Banach space setting to give short direct proofs of some results in [4], results involving the existence of an invariant mean on the weakly almost periodic functions on S or on the almost periodic functions on S. An ergodic theorem of Lloyd [16] is generalized, and a number of examples are presented.