Banach ideals of operators with applications

We present some results concerning the general theory of Banach ideals of operators and give several applications to Banach space theory. We give, in Section 3, new proofs of several recent results, as well as new operator characterizations of the $\text{L}$_p-spaces of Lindenstrauss and Pelczynski. In Section 4 we prove that the space of absolutely summing operators from E to F is reflexive if both E and F are reflexive and E has the approximation property. Section 5 concerns Hilbert spaces. In particular, we compute the relative projection constant of Hilbert spaces in L_p(μ)-spaces.     

Compactness in Vector-valued Banach Function Spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L_X^p, where X is a Banach space and 1≤ p<∞, and extend the result to vector-valued Banach function spaces E_X, where E is a Banach function space with order continuous norm. The author is supported by the ‘VIDI subsidie’ 639.032.201 in the ‘Vernieuwingsimpuls’ programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281.     

Compact operators in Banach lattices

Disjoint sequence methods from the theory of Riesz spaces are used to study compact operators on Banach lattices. A principal new result of the paper is that each positive map from a Banach latticeE to a Banach latticeF with compact majorant is itself compact provided the norms onE′ andF are order continuous.     

The distance between certainn-dimensional Banach spaces

IfE andF aren-dimensional Banach spaces, ifE has cotype 2, and if the ball ofF* has a small number of extreme points, then the Banach-Mazur distanced(E, F)≦C √nlogn. The techniques lead to the formally stronger result: IfE andF* have type 2 constantsa andb, respectively, thend(E, F)≦√n(a+b). IfE isn-dimensional, the identity map onE, when restricted to a large subspace ofE, factors through with normC √n. The authors’ work was supported in part, respectively, by NSF grant numbers MCS 78-02194, MCS 79-02489, and MCS 77-04174.     

Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds

Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x ₀)⊂E→F be C¹ nonlinear map, whereU (x ₀) is an open set containing pointx ₀∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x ₀) near x₀,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced calculus are established, including a theorem for C¹ Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives rise to a generalized principle for constructing Banach submanifolds.     

Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Let E be a real Banach space with property (α) and let W _Γ be an E-valued Brownian motion with distribution Γ. We show that a function is stochastically integrable with respect to W _Γ if and only if Γ-almost all orbits Ψx are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of Γ-measurable linear extensions of γ-radonifying operators with values in spaces of γ-radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution equations driven by an E-valued Brownian motion. The first named author gratefully acknowledges the support by a ‘VIDI subsidie’ in the ‘Vernieuwingsimpuls’ programme of The Netherlands Organization for Scientific Research (NWO) and the Research Training Network HPRN-CT-2002–00281. The second named author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1–1).     

An ergodic theorem on Banach lattices

E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf _n=ϕ means that the infimum of |f _n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA _n=(1/n)(I+T+...+T ⁿ⁻¹). Without an additional assumption onE, the “truncated limit” TLA _nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE ₊ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE ₊, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β. Research of this author is partially supported by NSERC Grant A3974. Research of this author is partially supported by NSF Grant 8301619.     

Embedding and Maximal Regular Differential Operators in Sobolev-Lions Spaces

This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.     

An example of a universal Banach space

We construct a separable reflexive Banach spaceX which is complementably universal for all finite dimensional Banach spaces. By this we mean: for every finite dimensional Banach spaceE there is isometric embeddingi:E→X such that there exists a projectionP: → _→ ^onto with ‖P‖=1.     

A Rank Theorem of Operators between Banach Spaces

Suppose that E and F are two Banach spaces and that B(E, F) is the space of all bounded linear operators from E to F. Let T ₀∈B(E, F) with a generalized inverse T ₀ ⁺∈B(F, E). This paper shows that, for every T∈B(E, F) with ‖T ₀ ⁺ (T−T ₀)‖<1, B ≡ (I + T ₀ ⁺(T−T ₀))⁻¹ T ₀ ⁺ is a generalized inverse of T if and only if (I−T ₀ ⁺ T ₀)N(T) = N(T ₀), where N(·) stands for the null space of the operator inside the parenthesis. This result improves a useful theorem of Nashed and Cheng and further shows that a lemma given by Nashed and Cheng is valid in the case where T ₀ is a semi-Fredholm operator but not in general.     

Singular integrals associated to the Laplacian on the affine groupax+b

We consider singular integral operators of the form (a)Z ₁L⁻¹Z₂, (b)Z ₁Z₂L⁻¹, and (c)L ⁻¹Z₁Z₂, whereZ ₁ andZ ₂ are nonzero right-invariant vector fields, andL is theL ²-closure of a canonical Laplacian. The operators (a) are shown to be bounded onL ^p for allp∈(1, ∞) and of weak type (1, 1), whereas all of the operators in (b) and (c) are not of weak type (p, p) for anyp∈[1, ∞). Research supported by the Australian Research Council. Research carried out as a National Research Fellow.     

Embeddings ofC(Δ) andL 1[0, 1] in Banach lattices

It is proved that ifE is a separable Banach lattice withE′ weakly sequentially complete,F is a Banach space andT:E→F is a bounded linear operator withT′F′ non-separable, then there is a subspaceG ofE, isomorphic toC(Δ), such thatT _G is an isomorphism, whereC(Δ) denotes the space of continuous real valued functions on the Cantor discontinuum. This generalizes an earlier result of the second-named author. A number of conditions are proved equivalent for a Banach latticeE to contain a subspace order isomorphic toC(Δ). Among them are the following:L ¹ is lattice isomorphic to a sublattice ofE′;C(Δ)′ is lattice isomorphic to a sublattice ofE′; E contains an order bounded sequence with no weak Cauchy subsequence;E has a separable closed sublatticeF such thatF′ does not have a weak order unit. The research of both authors was partially supported by the National Science Foundation, NSF Grant No MPS 71-02839 A04.     

Remarks on Banach spaces of compact operators

Let E and F be Banach spaces. We generalize several known results concerning the nature of the compact operators K(E, F) as a subspace of the bounded linear operators L(E, F). The main results are: (1) If E is a c₀ or l_p (1 < p < ∞) direct sum of a family of finite dimensional Banach spaces, then each bounded linear functional on K(E) admits a unique norm preserving extension to L(E). (2) If F has the bounded approximation property there is an isomorphism of $\text{L(E, F)}\text{into}\text{K(E, F)}{}^7$ such that its restriction to K(E, F) is the canonical injection. (3) If E is infinite dimensional and if F contains a complemented copy of c₀, K(E, F) is not complemented in L(E, F).     

Products and factors of Banach function spaces

Given two Banach function spaces we study the pointwise product space E · F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E · M(E, F) = F, where M(E, F) denotes the space of multiplication operators from E into F.     

B-convexity of the space of 2-summing operators

If π₂ (E, F), the class of 2-summing operators fromE intoF, isB-convex, then bothE andF have cotypeq for allq>2. On the other hand, ifE is super-reflexive and has cotype 2, then π₂(l ₂,E) is super-reflexive.     

On strongly-non-reflexive groups

A strong negative answer is given to the old question of whether every dual group is reflexive. Using ◊_ω1 a groupA is constructed so thatA, A*, A**, andA*** are weakly ω₁-separable groups of cardinalityω ₁ andA* is not isomorphic toA***. Research partially supported by NSF Grant No. DMS-8400451. Research partially supported by NSERC Grant No. A8948.     

Three classes of smooth Banach submanifolds in B(E,F)

Let E, F be two Banach spaces, and B(E, F), Φ(E, F), SΦ(E, F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result Let ∑ be any one of the following sets {T ∈ Φ(E, F) IndexT =const, and dim N(T) = const.}, {T ∈ SΦ(E, F) either dim N(T) = const. ＜ ∞ or codim R(T) = const.＜ ∞} and {T ∈ R(E, F) RankT =const.＜∞}. Then ∑ is a smooth submanifold of B(E, F) with the tangent space TA∑ = {B ∈ B(E,F) BN(A) (∪) R(A)} for any A ∈ ∑. The result is available for the further application to Thom's famous results on the transversility and the study of the infinite dimensional geometry.     

Almost everywhere convergence of Banach space-valued Vilenkin-Fourier series

The duality between martingale Hardy and BMO spaces is generalized for Banach space valued martingales. It is proved that if X is a UMD Banach space and f ∈ L _p(X) for some 1 < p < ∞ then the Vilenkin-Fourier series of f converges to f almost everywhere in X norm, which is the extension of Carleson’s result. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No. M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No. T043769, T047128, T047132.     

The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.