The spherical Bochner theorem on semisimple Lie groups

Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by $\text{C}$^p(G)(0<p?2), (ii) the K-biinvariant elements in $\text{C}$^p(G) by $\text{I}$^p(G), (iii) the positive definite (zonal) spherical functions by $\text{P}$, and (iv) the spherical transform on $\text{C}$^p(G) by ? → $\text{}\text{?}$gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto $\text{C}$^l(G), (ii) there exists a unique measure μ of polynomial growth on $\text{P}$ such that T[ψ]=∫pψdμ for all ψ?I¹(G) (iii) all measures μ of polynomial growth on $\text{P}$ are obtained in this way, and (iv) T may be extended to a particular $\text{I}$^p(G) space (1 ? p ? 2) if and only if the support of μ lies in a certain easily defined subset of $\text{P}$. These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.     

Productive local properties of function spaces

We characterize the spaces X for which the space Cp(X) of real valued continuous functions with the topology of pointwise convergence has local properties related to the preservation of countable tightness or the Fréchet property in products. In particular, we use the methods developed to construct an uncountable subset W of the real line such that the product of Cp(W) with any strongly Fréchet space is Fréchet. The example resolves an open question.     

Domain representability of certain function spaces

Let Cp(X) be the space of all continuous real-valued functions on a space X, with the topology of pointwise convergence. In this paper we show that Cp(X) is not domain representable unless X is discrete for a class of spaces that includes all pseudo-radial spaces and all generalized ordered spaces. This is a first step toward our conjecture that if X is completely regular, then Cp(X) is domain representable if and only if X is discrete. In addition, we show that if X is completely regular and pseudonormal, then in the function space Cp(X), Oxtoby's pseudocompleteness, strong Choquet completeness, and weak Choquet completeness are all equivalent to the statement “every countable subset of X is closed”.     

Weakly K-analytic spaces and the three-space property for analyticity

Let (E,E^′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E^′) to stronger ones in the frame of (E,E^′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.     

A grothendieck factorization theorem on 2-convex schatten spaces

We prove that for every bounded linear operatorT:C ^2p →H(1≤p<∞,H is a Hilbert space,C ^{2 p} p is the Schatten space) there exists a continuous linear formf onC ^p such thatf≥0, ‖f‖(C _C ^p)*=1 and $$\forall x \in C^{2p} , \left\| {T(x)} \right\| \leqslant 2\sqrt 2 \left\| T \right\|< f\frac{{x * x + xx*}}{2} > 1/2$$ . Forp=∞ this non-commutative analogue of Grothendieck’s theorem was first proved by G. Pisier. In the above statement the Schatten spaceC ^2p can be replaced byE _E ² whereE ⁽²⁾ is the 2-convexification of the symmetric sequence spaceE, andf is a continuous linear form onC _E. The statement can also be extended toL _E{(su2)}(M, τ) whereM is a Von Neumann algebra,τ a trace onM, E a symmetric function space.     

Optimal recovery from a finite set in Banach spaces of entire functions

We develop an approach to the problem of optimal recovery of continuous linear functionals in Banach spaces through information on a finite number of given functionals. The results obtained are applied to the problem of the best analytic continuation from a finite set in the complex space Cn, n?1, for classes of entire functions of exponential type which belong to the space Lp, 1<p<∞, on the real subspace of Cn. These latter are known as Wiener classes.     

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.     

Smoothness and duality in Lp(E, μ)

Let (T, Σ, μ) be a measure space, E a Banach space, and L_p(E, μ) the Lebesque-Bochner function spaces for 1 < p < ∞. It is shown that L_p(E, μ) is smooth if and only if E is smooth. From this result a Radon-Nikodym theorem for conjugates of smooth Banach spaces is established, and thus a general geometric condition on E sufficient to ensure that $\text{L}{}_{\mathrm{p}}\text{(E, μ)}{}^1\text{? L}{}_{\mathrm{q}}\text{(E}{}^1\text{, μ)}$ for all p, 1 < p < ∞. Alternate proofs of certain known results concerning the duals of L_p(E, μ) spaces are provided.     

Spherical functions on Euclidean space

We study spherical functions on Euclidean spaces from the viewpoint of Riemannian symmetric spaces. Here the Euclidean space En=G/K where G is the semidirect product Rn⋅K of the translation group with a closed subgroup K of the orthogonal group O(n). We give exact parameterizations of the space of (G,K)—spherical functions by a certain affine algebraic variety, and of the positive definite ones by a real form of that variety. We give exact formulae for the spherical functions in the case where K is transitive on the unit sphere in En.     

Orthogonally additive polynomials on spaces of continuous functions

We show that, for every orthogonally additive homogeneous polynomial P on a space of continuous functions C(K) with values in a Banach space Y, there exists a linear operator S:C(K)→Y such that P(f)=S(fn). This is the C(K) version of a related result of Sundaresam for polynomials on Lp spaces.     

Compact and strictly singular operators on Orlicz spaces

If 0 < p < 1 andT: L_p(0,1) →E is a continuous linear operator into a topological vector space, there is an infinite-dimensional subspaceX ofL _p on whichT is an isomorphism; thus there are no compact operators onL _p . Results of this type are proved for general non-locally convex Orlicz spaces.     

A metrizable X with Cp(X) not homeomorphic to Cp(X) × Cp(X)

We give an example of an infinite metrizable space X such that the space C_p(X), of continuous real-valued functions on X endowed with the pointwise topology, is not homeomorphic to its own square C_p(X) × C_p(X). The space X is a zero-dimensional subspace of the real line. Our result answers a long-standing open question in the theory of function spaces posed by A. V. Arhangel’skii.     

Coincidence of the Isbell and fine Isbell topologies

Let C(X,Y) be the set of all continuous functions from a topological space X into a topological space Y. We find conditions on X that make the Isbell and fine Isbell topologies on C(X,Y) equal for all Y. For zero-dimensional spaces X, we show there is a space Z such that the coincidence of the Isbell and fine Isbell topologies on C(X,Z) implies the coincidence on C(X,Y) for all Y. We then consider the question of when the Isbell and fine Isbell topologies coincide on the set of continuous real-valued functions. Our results are similar to results established for consonant spaces.     

On solvability of two-dimensional Lp-Minkowski problem

Given p≠0 and a positive continuous function g, with g(x+T)=g(x), for some 0<T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional has a minimizer in the class of positive functions u∈C¹(R) for which u(x+T)=u(x) for all x∈R. This minimizer is used to prove the existence of a positive periodic solution y∈C²(R) of two-dimensional L_p-Minkowski problem y^1−p(x)(y″(x)+y(x))=g(x), where p∉{0,2}.     

Point spectra of Cesàro matrices

For all positive a the point spectrum of the (C, α) matrix is determined, where the matrix is regarded as an operator on certain Banach sequence spaces. In particular the point spectrum is obtained in the spaces I_p(X), with 1<p≤∞, where X is a Banach space.     

Coarse version of the Banach-Stone theorem

We show that if there exists a Lipschitz homeomorphism T between the nets in the Banach spaces C(X) and C(Y) of continuous real valued functions on compact spaces X and Y, then the spaces X and Y are homeomorphic provided . By l(T) and l(T−1) we denote the Lipschitz constants of the maps T and T−1. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is . We also estimate the distance of the map T from the isometry of the spaces C(X) and C(Y).     

Hyperreflexivity and operator ideals

Suppose (B,β) is an operator ideal, and A is a linear space of operators between Banach spaces X and Y. Modifying the classical notion of hyperreflexivity, we say that A is called B-hyperreflexive if there exists a constant C such that, for any T∈B(X,Y) with α=supβ(qTi)<∞ (the supremum runs over all isometric embeddings i into X, and all quotient maps of Y, satisfying qAi=0), there exists a∈A, for which β(T−a)?Cα. In this paper, we give examples of B-hyperreflexive spaces, as well as of spaces failing this property. In the last section, we apply SE-hyperreflexivity of operator algebras (SE is a regular symmetrically normed operator ideal) to constructing operator spaces with prescribed families of completely bounded maps.     

Reflexive operators with domain in Köthe spaces

It is proved that if a K?the space λ₁(A) is distinguished and E is an arbitrary Fréchet space then every reflexive map T: λ₁(A)→E (i.e., T maps bounded sets into relatively weakly compact ones) factorizes through a reflexive Fréchet space. An analogous result is proved for Montel maps (i.e., which map bounded sets into relatively compact ones). The result is a consequence of the fact proved also in this paper that, for a distinguished λ₁(A) space, the spaces of reflexive maps R(λ₁(A), C(K)) and of Montel maps M(λ₁(A), C(K)) are the Mackey completions of the spaces of weakly compact and compact maps, respectively. Consequences for spaces of vector-valued (weakly) continuous functions are also obtained. Received: 24 November 1997 / Revised version: 14 May 1998     

Most of the successive approximations do converge

Let T be a locally compact Hausdorff space and E a Banach space. Let K(T, E) be the set of all continuous E-valued functions on T with compact support. We consider the representation of the second dual of K(T, E) when K(T, E) is normed with the usual sup norm. We demonstrate that an operator in the second dual of K(T, E) is, in a certain sense, approximable by an integral when computed over a certain subset of the dual of K(T, E).