Interpolation of uniformly convex Banach spaces

Journal of Computational and Applied Mathematics, 1997

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.

Arkiv för matematik, 1989

Journal of Functional Analysis, 1986

We interpolate in the complex method some real-intermediate quasi-Banach spaces. This enables us for example to get in a unilied way complex interpolation of H,, spaces 0 <pO <pr 4 co, from the real interpolation results. The H, spaces could be the standard ones as well as weighted H, spaces, H,, spaces on product domains, etc. (' 1986 Academic Press. Inc An extension of the A. P. Calderon method of complex interpolation to quasi-Banach spaces was first considered by N. M. Riviere in [ 111. The main obstacle to a fully successful theory is the failure of the maximum principle for functions assuming values in a quasi-Banach space; see . We should point out that for the real interpolation method the situation is quite different; most central theorems from the Banach space setting hold in the quasi-Banach, and even in a more general situation.

We recall that the fundamental theorem of complex interpolation is the Boundedness Theorem: If, for j = 0, 1, a linear operator T is a bounded map from the Banach space X j to the Banach space Y j then, for each θ, 0 < θ < 1, T is a bounded map between the complex interpolation spaces [X 0 , X 1 ] θ and [Y 0 , Y 1 ] θ. Alberto Calderón, in his foundational presentation of this material fifty-one years ago [4], also proved the following companion result: Compactness Theorem/Question: Furthermore in some cases, if T is also a compact map from X 0 to Y 0 , then, for each θ, T is a compact map from [X 0 , X 1 ] θ to [Y 0 , Y 1 ] θ. The fundamental question of exactly which cases could be covered by such a result was not resolved then, and is still not resolved. In a previous paper [10] we surveyed several of the partial answers which have been obtained to this question, with particular emphasis on the work of Nigel Kalton in a joint paper [8] with one of us. This is a preliminary version of a set of lecture notes which will be a sequel to [10]. In them, for the most part, we will amplify upon various technical details of the contents of [8]. For example we plan to give a more explicit explanation of why the positive answer in [8] to the above question when (X 0 , X 1) is a couple of lattices holds without any restriction on those lattices, and we also plan to provide more detailed versions of some of the other proofs in that paper. The main purpose of this preliminary version is to present two apparently new small results, pointing out a previously unnoticed particular case where the answer to the above question is affirmative. As our title suggests, this and future versions of these notes are intended to be more accessible to graduate students than a usual research article.

Transactions of the American Mathematical Society, 1984

The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville’s theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.

Mathematical Notes, 2014

Let (X 0 , X 1 ) and (Y 0 , Y 1 ) be complex Banach couples and assume that X 1 ⊆ X 0 with norms satisfying x X 0 ≤ c x X 1 for some c > 0. For any 0 < θ < 1, denote by X θ = [X 0 , X 1 ] θ and Y θ = [Y 0 , Y 1 ] θ the complex interpolation spaces and by B(r, X θ ), 0 ≤ θ ≤ 1, the open ball of radius r > 0 in X θ , centered at zero. Then for any analytic map Φ : B(r, X 0 ) → Y 0 + Y 1 such that Φ : B(r, X 0 ) → Y 0 and Φ : B(c −1 r, X 1 ) → Y 1 are continuous and bounded by constants M 0 and M 1 , respectively, the restriction of Φ to B(c −θ r, X θ ), 0 < θ < 1, is shown to be a map with values in Y θ which is analytic and bounded by M 1−θ 0 M θ 1 .

Journal of Approximation Theory, 1969

Commentationes Mathematicae Universitatis Carolinae, 2016

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2000

Lorentz and Shimogaki [2] have characterized those pairs of Lorentz A spaces which satisfy the interpolation property with respect to two other pairs of A spaces. Their proof is long and technical and does not easily admit to generalization. In this paper we present a short proof of this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics will be reported on elsewhere. The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable functions ƒ on the interval (0, /) for which the norm is finite, where </> is an integrable, positive, decreasing function on (0, /) and/* (the decreasing rearrangement of |/|) is the almost-everywhere unique, positive, decreasing function which is equimeasurable with \f\. A pair of spaces (A^, A v) is called an interpolation pair for the two pairs (A^, A Vl) and (A^2, A V2) if each linear operator which is bounded from A^ to A v (both /== 1, 2) has a unique extension to a bounded operator from A^ to A v. THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition that (A^, A w) be an interpolation pair for (A^, A Vi) and (A^2, A V2) is that there exist a constant A independent of s and t so that (*) ^(0/0(5) ^ A max(TO/^(a)) t=1.2 holds, where O 00=ƒ S <j>{r) dr,-" , VaC'Wo Y a (r) dr.

2007

We study Carathéodory-Herglotz functions whose values are continuous operators from a locally convex topological space which admits the factorization property into its conjugate dual space. We show how this case can be reduced to the case of functions whose values are bounded operators from a Hilbert space into itself. 2000 Mathematics Subject Classification. 47Bxx,46A03. Key words and phrases. Positive kernels, locally convex spaces. 1 We used the terminology Carathéodory function in [3].