Stability in interpolation of families of Banach spaces

Journal of Functional Analysis, 2002

Let (A 0 , A 1 ) and (B 0 , B 1 ) be two interpolation couples and let T: (A 0 , A 1 ) W (B 0 , B 1 ) be a K-quasilinear operator. The boundedness of the operator from A 0 to B 0 implies K(t, Ta; B 0 , B 1 ) [ M 0 ||a|| A0 and the boundedness of the operator from A 1 to B 1 implies K(t, Ta; B 1 , B 0 ) [ M 1 ||a|| A1 , a ¥ A 0 5 A 1 . We consider perturbations of these two inequalities in the form

The Quarterly Journal of Mathematics, 1999

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.

Journal of the Institute of Mathematics of Jussieu, 2020

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

Proceedings of the American Mathematical Society, 1995

Let X and its dual X* be uniformly convex Banach spaces, D an open and bounded subset of X, Ta continuous and pseudo-contractive mapping defined on cl(D) and taking values in X. If T satisfies the following condition: there exists z £ D such that \\z-Tz\\ < \\x-Tx\\ for all x on the boundary of D , then the trajectory t-* zt £ D, t G [0, 1), defined by Z(= tT(z,) + (1-t)z is continuous and converges strongly to a fixed point of T as t-> 1- .

arXiv: Functional Analysis, 2020

We prove the stability of isomorphisms between Banach spaces generated by interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also the so-called $\pm$ or $G_1$ and $G_2$ methods defined by Peetre and Gustavsson-Peetre. This result is used to show the existence of solution of certain operator analytic equation. A by product of these results is a more general variant of the Albrecht-Muller result which states that the interpolated isomorphisms satisfy uniqueness-of-inverses between interpolation spaces. We show applications for positive operators between Calderon function lattices. We also derive connections between the spectrum of interpolated operators.

Indagationes Mathematicae (Proceedings), 1974

Let A be a meromorphic function with values in the space of bounded linear operators between two Banach spaces X and Y, and assume that the coefficients of the prinoipal part of the Laurent expansion of A at a certain point & are degenerate operators. In this paper it is shown that under rather general hypotheses the null spaces (resp. ranges) of A(1) converge in the gap topology to a certain subspace of X (reap. Y) as il approaches &. Further, under slightly stronger conditions, the null spaces (reap. ranges) of A(A) have E fixed complementary subspace in X (resp. Y) for all rl in some deleted neighbourhood of k. The hypotheses of these stability theorems are fulfilled if A is Fredholm at & or haa values in the set of degenerate operators.

Integral Equations and Operator Theory, 1981

Let R be a bounded domain in the complex plane bounded by n + 1 nonintersecting analytic Jordan curves, let E, F, and G be flat unitary vector bundles (in the sense of Abrahamse and Douglas) and let O: F~ G and ~: E ~ G be bounded analytic bundle maps. A condition is given for the existence of a bounded analytic map D: E---" F such that OD = ~, together with an estimate for I]D[~o. An interesting special case is the case where E = G and • ---I E , for which the condition involves a uniform lower bound for a class of Toeplitz operators over R, all of which are induced (formally) by the N bundle map @ 0 (N = rank E). When interpreted for a finite column of anal lytic scalar functions, this special case gives quantitative information on the corona theorem for R. The main tool is the Sz.Nagy-Foias commutant lifting theorem for regions R recently obtained by the author.

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 .

2015

Using the results in papers [2] and [3] in this paper we prove the existence of the interpolating spline-function by the null space dimension of operators A and T. 1. Introduction. Let X,Y,Z be Banach spaces. Suppose A is a bounded linear operator of X into Z and T is bounded linear operator of X into Y. The null space and the rang of operator A will be denoted by N (A) and R(A), respectively. Let R(A) = Z. For a fixed element z ∈ Z we write by