Spaces of compact operators and their dual spaces

Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒ^w' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T _n)⊂K(X, Y) such thatT=ω′−lim_nT_n and let . We show that is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in . This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces. Financial support from the Potchefstroom University and Maseno University is greatly acknowledged. Financial support from the NRF and Potchefstroom University is greatly acknowledged.     

On james’s paper “separable conjugate spaces”

For every separable Banach spaceX there is a Banach spaceY with a separable dual such thatY ⊕X* ≈Y**. There is also a separable spaceZ so thatZ**/JZ is isomorphic toX.     

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

On normal and compact operators on Banach spaces

For some normal operators (T=H+iK) on a Banach spaceX we study the dual space of the Banach algebraA (H, K) assuming thatX* is weakly complete and we study the decompositionX=Ker (T) ⊕ (TX)⁻ for spacesX ⊅c ₀.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT ⁻ onY, and the spectrum ofT ⁻ is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT ⁻ onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

A property of strictly singular one-to-one operators

We prove that ifT is a strictly singular one-to-one operator defined on an infinite dimensional Banach spaceX, then for every infinite dimensional subspaceY ofX there exists an infinite dimensional subspaceZ ofX such thatZ∩Y is infinite dimensional,Z contains orbits ofT of every finite length and the restriction ofT toZ is a compact operator. The research was partially supported by NSF.     

A non-reflexive Grothendieck space that does not containl ∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)^*, every weak^* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol _∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl _∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.     

On dualL 1-spaces and injective bidual Banach spaces

In a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l ¹(τ) embeds in a Banach spaceX if and only ifL ¹([0, 1]^τ) embeds inX ^*. An extension (to a rather wider class of cardinals) of the basic lemma of that paper is here applied so as to yield an affirmative answer to a question posed by Rosenthal concerning dual ℒ₁-spaces. It is shown that ifZ ^* is a dual Banach space, isomorphic to a complemented subspace of anL ¹-space, and κ is the density character ofZ ^*, thenl ¹(κ) embeds inZ ^*. A corollary of this result is that every injective bidual Banach space is isomorphic tol ^∞(κ) for some κ. The second part of this article is devoted to an example, constructed using the continuum hypothesis, of a compact spaceS which carries a homogeneous measure of type ω₁, but which is such thatl ¹(ω₁) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω₁. The dual space ℰ(S)^* is isometric to , and is a member of a new isomorphism class of dualL ¹-spaces.     

L 1 (μ, X) as a constrained subspace of its bidual

In this note we consider the property of being constrained in the bidual, for the space of Bochner integrable functions. For a Banach spaceX having the Radon-Nikodym property and constrained in its bidual and forY ⊂ X, under a natural assumption onY, we show thatL ¹ (μ, X/Y) is constrained in its bidual andL ¹ (μ, Y) is a proximinal subspace ofL ¹(μ, X). As an application of these results, we show that, ifL ¹(μ, X) admits generalized centers for finite sets and ifY ⊂ X is reflexive, thenL ¹ μ, X/Y) also admits generalized centers for finite sets.     

Injective Banach spaces of typeC(T)

A Banach spaceX is aP _λ-space if wheneverX is isometrically embedded in another Banach spaceY there is a projection ofY ontoX with norm at most λ.C(T) denotes the Banach space of continuous real-valued functions on the compact Hausdorff spaceT. T satisfies the countable chain condition (CCC) if every family of disjoint non-empty open sets inT is countable.T is extremally disconnected if the closure of every open set inT is open. The main result is that ifT satisfies the CCC andC(T) is aP _λ-space, thenT is the union of an open dense extremally disconnected subset and a complementary closed setT _Asuch thatC(T_A) is aP _λ?1-space.     

On banach spaces which containl 1(τ) and types of measures on compact spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL ¹. Let τ be a regular cardinal satisfying the hypothesis that κ^ω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]^τ. 2) A Banach spaceX has a subspace isomorphic tol ¹(τ) if and only ifX ^∗ has a subspace isomorphic toL ¹({0, 1}^τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl ¹ does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl ¹ (ω₁), while the unit ball ofX ^∗ is not weakly^∗ sequentially compact.     

The problem of envelopes for Banach spaces

LetX be a Banach space. A Banach spaceY is an envelope ofX if (1)Y is finitely representable inX; (2) any Banach spaceZ finitely representable inX and of density character not exceeding that ofY is isometric to a subspace ofY. Lindenstrauss and Pelczynski have asked whether any separable Banach space has a separable envelope. We give a negative answer to this question by showing the existence of a Banach space isomorphic tol ₂, which has no separable envelope. A weaker positive result holds: any separable Banach space has an envelope of density character ≦ℵ₁ (assuming the continuum hypothesis).     

Extremely non-complex Banach spaces

A Banach space X is said to be an extremely non-complex space if the norm equality ∥Id +T ²∥ = 1+∥T ²∥ holds for every bounded linear operator T on X. We show that every extremely non-complex Banach space has positive numerical index, it does not have an unconditional basis and that the infimum of diameters of the slices of its unit ball is positive.     

Minimal Extensions in Tensor Product Spaces

LetX,Ybe two separable Banach spaces and letVXandWYbe finite dimensional subspaces. Suppose thatVSX,WZYand letM (S, V),N (Z, W). We will prove that ifαis a reasonable, uniform crossnorm onXYthenλ_MN(V_αW,X_αY)=λ_M(V, X) λ_N(W, Y).Here for any Banach spaceX,VSXandM (S, V)

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Also some applications of the above mentioned result will be presented.     

Sobolev embeddings into BMO, VMO, andL ∞

LetX be a rearrangement-invariant Banach function space onR ⁿ and letV ¹ X be the Sobolev space of functions whose gradient belongs toX. We give necessary and sufficient conditions onX under whichV ¹ X is continuously embedded into BMO or intoL _∞. In particular, we show thatL _{n, ∞} is the largest rearrangement-invariant spaceX such thatV ¹ X is continuously embedded into BMO and, similarly,L _{n, 1} is the largest rearrangement-invariant spaceX such thatV ¹ X is continuously embedded intoL _∞. We further show thatV ¹ X is a subset of VMO if and only if every function fromX has an absolutely continuous norm inL _{n, ∞} . A compact inclusion ofV ¹ X intoC ⁰ is characterized as well.     

<Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>-Singular and <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript>-singular operators between vector-valued Banach lattices

Given an operator T : X → Y between Banach spaces, and a Banach lattice E consisting of measurable functions, we consider the point-wise extension of the operator to the vector-valued Banach lattices T _E : E(X) → E(Y) given by T _E (f)(ω) = T(f(ω)). It is proved that for any Banach lattice E which does not contain c ₀, the operator T is an isomorphism on a subspace isomorphic to c ₀ if and only if so is T _E . An analogous result for invertible operators on subspaces isomorphic to ℓ ₁ is also given.     

C-semigroups and strongly continuous semigroups

We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA| _Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖_[Im(C)]≡‖C ⁻¹ x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B| _X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true. We construct fractional powers of generators of boundedC-semigroups. We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.     

Function spaces not containing ℓ<Subscript>1</Subscript>

For Ω bounded and open subset of andX a reflexive Banach space with 1-symmetric basis, the function spaceJF _X (Ω) is defined. This class of spaces includes the classical James function space. Every member of this class is separable and has non-separable dual. We provide a proof of topological nature thatJF _X (Ω) does not contain an isomorphic copy of ℓ₁. We also investigate the structure of these spaces and their duals.