On the reflexivity of $$\mathcal {P}_{w}(^{n}E;F)$$

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.     

Persistence of Banach lattices under nonlinear order isomorphisms

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection \(T: E\rightarrow F\) such that \(x\ge y\) if and only if \(Tx \ge Ty\) for all \(x,y \in E\). We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to \(c_{0}\), and for Banach lattices that contain a closed sublattice order isomorphic to \(c_{0}\).     

Banach–Stone Theorems for maps preserving common zeros

Let X and Y be completely regular spaces and E and F be Hausdorff topological vector spaces. We call a linear map T from a subspace of C(X, E) into C(Y, F) a Banach–Stone map if it has the form T f (y) =  S _y (f (h(y))) for a family of linear operators S _y : E → F, \({y \in Y}\) , and a function h: Y → X. In this paper, we consider maps having the property:

$\bigcap^{k}_{i=1}Z(f_{i}) \neq\emptyset \iff \bigcap^{k}_{i=1}Z(Tf_{i})\neq\emptyset , \quad({\rm Z}) $

where Z(f) =  {f =  0}. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including C ^∞), as Banach–Stone maps. In particular, we confirm a conjecture of Ercan and Önal: Suppose that X and Y are realcompact spaces and E and F are Hausdorff topological vector lattices (respectively, C ^*-algebras). Let T: C(X, E) → C(Y, F) be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that

$ Z(f) \neq\emptyset\iff Z(Tf) \neq\emptyset. $

Then X is homeomorphic to Y and E is lattice isomorphic (respectively, C ^*-isomorphic) to F. Some results concerning the continuity of T are also obtained.

     

Strictly singular operators in pairs of <Emphasis Type="Italic">L</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript> space

Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q ? E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from L_p to L_q is found. There exists a strictly singular but not superstrictly singular operator on L_p, provided that p ≠ 2.     

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L₁(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.     

Strict <Emphasis Type="Italic">K</Emphasis>-monotonicity and <Emphasis Type="Italic">K</Emphasis>-order continuity in symmetric spaces

This paper is devoted to strict K-monotonicity and K-order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K-order continuity in a symmetric space E on \([0,\infty )\) implies that the embedding \(E\hookrightarrow {L^1}[0,\infty )\) does not hold. We present a complete characterization of an equivalent condition to K-order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K-monotonicity and K-order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec–Klee property for global convergence in measure in a symmetric space E.     

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain \(\hat \otimes\)-algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ?: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H ⁿ(?): H ⁿ (x) → H ⁿ (y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective \(\hat \otimes\)-algebras: the tensor algebra E \(\hat \otimes\) F generated by the duality (E,F,<·,·>) for nuclear Fréchet spaces E and F or for nuclear DF-spaces E and F; nuclear biprojective Köthe algebras λ(P) which are Fréchet spaces or DF-spaces; the algebra of distributions ε^*(G) on a compact Lie group G.     

Banach space structure of weighted Fock-Sobolev spaces

We discuss the Banach space structure of the fractional order weighted Fock-Sobolev spaces F _α,s ^p , mainly include giving some growth estimates for Fock-Sobolev functions and approximating them by a sequence of ‘nice’ functions in two different ways.     

Bochner integrals in ordered vector spaces

We present a natural way to cover an Archimedean directed ordered vector space E by Banach spaces and extend the notion of Bochner integrability to functions with values in E. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map.     

On the “function” and “lattice” definitions of a narrow operator

We prove that the function and lattice definitions of a narrow operator defined on a Köthe Banach space E on a finite atomless measure space \((\Omega , \Sigma , \mu )\) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013).     

Numerically Hypercyclic Operators

An operator T acting on a normed space E is numerically hypercyclic if, for some \({(x, x^*)\in \Pi(E)}\), the numerical orbit \({\{x^*(T^n(x)):n\geq 0\}}\) is dense in \({\mathbb{C}}\). We prove that finite dimensional Banach spaces with dimension at least two support numerically hypercyclic operators. We also characterize the numerically hypercyclic weighted shifts on classical sequence spaces.     

Affine embeddings of Cantor sets and dimension of <Emphasis Type="Italic">αβ</Emphasis>-sets

Let E,F ? R^d be two self-similar sets, and suppose that F can be affinely embedded into E. Under the assumption that E is dust-like and has a small Hausdorff dimension, we prove the logarithmic commensurability between the contraction ratios of E and F. This gives a partial affirmative answer to Conjecture 1.2 in [9]. The proof is based on our study of the boxcounting dimension of a class of multi-rotation invariant sets on the unit circle, including the αβ-sets initially studied by Engelking and Katznelson.     

Weakly complex Einstein-Finsler vector bundle

Let (E, F) be a complex Finsler vector bundle over a compact Kähler manifold (M, g) with Kähler form Φ. We prove that if (E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou (1997), then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle (E, F) over a compact Kähler manifold (M, g) is necessarily Φ-semistable and (E, F) = (E₁, F₁) ? · · · ? (Ek; Fk); where F _j := F |E _j , and each (E _j , F _j ) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as (E, F).     

Invariant <Emphasis Type="Italic">f</Emphasis>-structures on naturally reductive homogeneous spaces

We study invariant metric f-structures on naturally reductive homogeneous spaces and establish their relation to generalized Hermitian geometry. We prove a series of criteria characterizing geometric and algebraic properties of important classes of metric f-structures: nearly Kähler, Hermitian, Kähler, and Killing structures. It is shown that canonical f-structures on homogeneous Φ-spaces of order k (homogeneous k-symmetric spaces) play remarkable part in this line of investigation. In particular, we present the final results concerning canonical f-structures on naturally reductive homogeneous Φ-spaces of order 4 and 5.     

The <Emphasis Type="Italic">p</Emphasis>-Gelfand–Phillips property in spaces of operators and Dunford–Pettis like sets

The p-Gelfand–Phillips property (1 \({\leq}\) p < ∞) is studied in spaces of operators. Dunford–Pettis type like sets are studied in Banach spaces. We discuss Banach spaces X with the property that every p-convergent operator T:X \({\rightarrow}\) Y is weakly compact, for every Banach space Y.     

Optimal Embeddings of Bessel-Potential-Type Spaces into Generalized Hölder Spaces Involving <Emphasis Type="Italic">k</Emphasis>-Modulus of Smoothness

We establish necessary and sufficient conditions for embeddings of Bessel potential spaces H ^σ X(IR ⁿ ) with order of smoothness σ?∈?(0, n), modelled upon rearrangement invariant Banach function spaces X(IR ⁿ ), into generalized Hölder spaces (involving k-modulus of smoothness). We apply our results to the case when X(IR ⁿ ) is the Lorentz-Karamata space \(L_{p,q;b}({{\rm I\kern-.17em R}}^n)\). In particular, we are able to characterize optimal embeddings of Bessel potential spaces \(H^{\sigma}L_{p,q;b}({{\rm I\kern-.17em R}}^n)\) into generalized Hölder spaces. Applications cover both superlimiting and limiting cases. We also show that our results yield new and sharp embeddings of Sobolev-Orlicz spaces W ^k?+?1 L ^n/k(logL) ^α (IR ⁿ ) and W ^k L ^n/k(logL) ^α (IR ⁿ ) into generalized Hölder spaces.     

Ideals of subseries convergence and <Emphasis Type="Italic">F</Emphasis>-spaces

Let X be an F-space and \({\boldsymbol x=(x_n)}\) be a sequence of vectors in X. Ideals \({\mathcal{C}(\boldsymbol x)}\) of subseries convergence are considered. In particular, we show that a characterization of the class of Banach spaces not containing c ₀ obtained by using the ideals \({\mathcal{C}(\boldsymbol x)}\) breaks down in every Fréchet space not isomorphic to a Banach space. On the other hand, the result can be extended to some F-spaces via the definition of a new class of F-spaces satisfying a stronger version of the condition (O) of Orlicz. A theorem discriminating between the finite and infinite dimensional case is obtained about the family \({\mathcal{C}(X)}\) of all ideals associated with the F-space X.     

Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane

In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of causal weighted composition operators on these spaces are related to each other and use it to show that an (unweighted) composition operator \(C_\varphi \) is bounded on a Zen space if and only if \(\varphi \) has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces.