Reiteration formulae for interpolation methods associated to polygons

We study spaces generated by applying the interpolation methods defined by a polygon Π to an N-tuple of real interpolation spaces with respect to a fixed Banach couple {X,Y}. We show that if the interior point (α,β) of the polygon does not lie in any diagonal of Π then the interpolation spaces coincide with sums and intersections of real interpolation spaces generated by {X,Y}. Applications are given to N-tuples formed by Lorentz function spaces and Besov spaces. Moreover, we show that results fail in general if (α,β) is in a diagonal.     

Interpolation of Limited and Weakly Compact Operators on Families of Banach Spaces

In this paper, we study how the limited and weakly compact properties of operators are preserved by interpolation of the real method for infinite families of Banach spaces introduced by Carro in Studia Math. 109 (1994). We apply these results to the case of Sparr, Fernández and Cobos–Peetre methods of interpolation for finite families.     

The Fundamental Function of Spaces Generated by Interpolation Methods Associated to Polygons

We consider the K- and J-spaces generated from an N-tuple of rearrangement-invariant function spaces by using the interpolation methods associated to polygons. We compute their fundamental functions and we characterize their associate spaces. Furthermore, an application is given to interpolation of N-tuples of Marcinkiewicz spaces.     

On interpolation of bilinear operators

In this paper we study interpolation of bilinear operators between products of Banach spaces generated by abstract methods of interpolation in the sense of Aronszajn and Gagliardo. A variant of bilinear interpolation theorem is proved for bilinear operators from corresponding weighted c₀ spaces into Banach spaces of non-trivial the periodic Fourier cotype. This result is then extended to the spaces generated by the well-known minimal and maximal methods of interpolation determined by quasi-concave functions. In the case when a maximal construction is generated by Hilbert spaces, we obtain a general variant of bilinear interpolation theorem. Combining this result with the abstract Grothendieck theorem of Pisier yields further results. The results are applied in deriving a bilinear interpolation theorem for Calderón-Lozanovsky, for Orlicz spaces and an embedding interpolation formula for (E,p)-summing operators.     

Hypercyclic tuples of operators and somewhere dense orbits

In this paper we prove that there are hypercyclic (n+1)-tuples of diagonal matrices on Cn and that there are no hypercyclic n-tuples of diagonalizable matrices on Cn. We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense.     

Asymptotic Behavior of Factorization and Projection Constants

This paper is concerned with estimates of important factorization constants that appear in Banach space theory. We prove upper bounds of the Hilbertian norm of projections on finite-dimensional spaces of interpolation spaces generated by certain abstract interpolation functors and show applications to Calderón–Lozanovskii spaces. We also prove estimates of the p-factorization norm and projection constants for finite-dimensional Banach lattices. We show as a consequence of our results that in a large class of n-dimensional Banach sequence lattices \(E_n\) the projection constants \(\lambda (E_n)\) satisfy \(\lim _{n\rightarrow \infty }\lambda (E_n)/\sqrt{n} = c\), where \(c=\sqrt{2/\pi }\) in the real case and \(c= \sqrt{\pi }/2\) in the complex case. Applications are given to vector-valued sequence spaces.     

Complex interpolation of Orlicz spaces with respect to a vector measure

We apply the Calderón interpolation methods to Orlicz and weakly Orlicz function spaces with respect to a Banach‐space‐valued measure defined on a σ‐algebra. The results we obtain generalize those in the case of Banach lattices of p‐integrable and weakly p‐integrable functions with respect to such a vector measure.     

Analytic mappings between LB-spaces and applications in infinite-dimensional Lie theory

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm (K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group ${\bigcup_{n\in\mathbb {N}}G_n}We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm (K, X) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group è_{n ? \mathbb N}G_n{\bigcup_{n\in\mathbb {N}}G_n} where the G _n are Banach Lie groups.     

Universal extrapolation spaces for \hbox {C}_{0}-semigroups

The classical theory of Sobolev towers allows for the construction of an infinite ascending chain of extrapolation spaces and an infinite descending chain of interpolation spaces associated with a given \(C_0\) -semigroup on a Banach space. In this note we first generalize the latter to the case of a strongly continuous and exponentially equicontinuous semigroup on a complete locally convex space. As a new concept—even for \(C_0\) -semigroups on Banach spaces—we then define a universal extrapolation space as the completion of the inductive limit of the ascending chain. Under mild assumptions we show that the semigroup extends to this space and that it is generated by an automorphism of the latter. Dually, we define a universal interpolation space as the projective limit of the descending chain. We show that the restriction of the initial semigroup to this space is again a semigroup and always has an automorphism as generator.     

Minimal ideals of n-homogeneous polynomials on Banach spaces

The minimal kernel of a p-Banach ideal of n-homogeneous polynomials between Banach spaces is defined as a composition ideal, characterized to be the smallest ideal which coincides with the given one on finite-dimensional spaces and represented through tensor products with appropriate norms.     

On interpolation of function spaces by methods defined by means of polygons

We describe the spaces obtained by applying the interpolation methods associated to polygons to N-tuples of weighted L^p-spaces, N-tuples of classical Lorentz spaces and some other N-tuples of function spaces.     

The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

Towards strong Banach property (T) for SL(3,ℝ)

We prove that SL(3, ?) has Strong Banach property (T) in Lafforgue’s sense with respect to the Banach spaces that are θ > 0 interpolation spaces (for the complex interpolation method) between an arbitrary Banach space and a Banach space with sufficiently good type and cotype. As a consequence, every action of SL(3, ?) or its lattices by affine isometries on such a Banach space X has a fixed point, and the expanders contructed from SL(3, ?) do not admit a coarse embedding into X. We also prove a quantitative decay of matrix coefficients (Howe-Moore property) for representations with small exponential growth of SL(3, ?) on X.     

Boundary conditions for linear manifolds. III. Infinite dimensional adjoints via solution spaces

A formula is given for the orthogonal complement of any vector subspace of l₂. Countably infinite adjoint subspaces in a Banach space are characterized via solution spaces. In particular, infinite dimensional self-adjoint subspaces in a reflexive Banach space are characterized via solution spaces, generalizing a result in Dunford and Schwartz [“Linear Operators, II,” Interscience, New York, 1963]. Applications are made to closed linear manifolds in l₂ ⊕ l₂ as well as infinite dimensional, generalized ordinary differential subspaces in a Hilbert space with the boundary conditions imposed on real sequences. The results are also expressed via solution spaces.     

Compact embeddings and proper mappings

Sufficient conditions are given to assert that a C¹-proper mapping defined on a compact embedding between Banach spaces over K has a fixed point. The proof of the result is based upon continuation methods.     

On certain Banach spaces in connection with interpolation theory

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper certain Banach spaces. A relation between these spaces and the space (C₀,S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.Finally, certain inequalities for the minimax series of a product are obtained which permit us to define a product over the space (C₀,S). With such a product (C₀,S) is a Banach Algebra.     

Banach spaces of universal disposition

In this paper we present a method to obtain Banach spaces of universal and almost-universal disposition with respect to a given class M of normed spaces. The method produces, among others, the only separable Banach space of almost-universal disposition with respect to the class F of finite-dimensional spaces (Gurari? space G); or the only, under CH, Banach space with density character the continuum which is of universal disposition with respect to the class S of separable spaces (Kubis space K). We moreover show that K is isomorphic to an ultrapower of the Gurari? space and that it is not isomorphic to a complemented subspace of any C(K)-space. Other properties of spaces of universal disposition are also studied: separable injectivity, partially automorphic character and uniqueness.     

Absolutely summing multipliers on abstract Hardy spaces

We study summing multipliers from Banach spaces of analytic functions on the unit disc of the complex plane to the complex Banach sequence lattices. The domain spaces are abstract variants of the classical Hardy spaces generated by the complex symmetric spaces. Applying interpolation methods, we prove the Hausdorff Young and Hardy-Littlewood type theorems. We show applications of these results to study summing multipliers from the Hardy-Orlicz spaces to the Orlicz sequence lattices. The obtained results extend the well-known results for the Hp spaces.