Indices defined by interpolation scales and applications

Mathematical Proceedings of the Cambridge Philosophical Society, 2004

We investigate inclusion indices for general function spaces, not necessarily symmetric. Using them, we estimate the grade of proximity between two spaces E → F when we have certain information on the inclusion. The results are based on ideas from interpolation theory.

The Quarterly Journal of Mathematics, 1999

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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.

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.

Studia Mathematica, 2000

We show that the numerical index of a c 0-, l 1-, or l ∞-sum of Banach spaces is the infimum numerical index of the summands. Moreover, we prove that the spaces C(K, X) and L 1 (µ, X) (K any compact Hausdorff space, µ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.

2016

Consider a Banach space valued measurable function $f$ and an operator $u$ from the space where {$f$} takes values. If $f $ is Pettis integrable, a classical result due to J. Diestel shows that composing it with $u$ gives a Bochner integrable function $u \circ f$ whenever $u$ is absolutely summing. In a previous work we have shown that a well-known interpolation technique for operator ideals allows to prove under some requirements that a composition of a $p$-Pettis integrable function with a $q$-summing operator provides an $r$-Bochner integrable function. In this paper a new abstract inclusion theorem for classes of {abstract} summing operators is shown and applied to the class of interpolated operator ideals. Together with the results of the {aforementioned} paper, it provides more results on the relation about the integrability of the function $u \circ f$ and the summability properties of $u$.

Banach Journal of Mathematical Analysis, 2011

Approximation spaces, in their many presentations, are well known mathematical objects and many authors have studied them for long time. They were introduced by Butzer and Scherer in 1968 and, independently, by Y. Brudnyi and N. Kruglyak in 1978, and popularized by Pietsch in his seminal paper of 1981. Pietsch was interested in the parallelism that exists between the theories of approximation spaces and interpolation spaces, so that he proved embedding, reiteration and representation results for approximation spaces. In particular, embedding results are a natural part of the theory since its inception. The main goal of this paper is to prove that, for certain classes of approximation schemes (X, {A n }) and sequence spaces S, if S 1 ⊂ S 2 ⊂ c 0 (with strict inclusions) then the approximation space A(X, S 1 , {A n }) is properly contained into A(X, S 2 , {A n }). We also initiate a study of strict inclusions between interpolation spaces, for Petree's real interpolation method.

Dissertationes Mathematicae, 2020

The aim of this paper is to study the numerical index with respect to an operator between Banach spaces. Given Banach spaces X and Y , and a norm-one operator G ∈ L(X, Y) (the space of all bounded linear operator from X to Y), the numerical index with respect to G, n G (X, Y), is the greatest constant k 0 such that k T inf δ>0 sup |y * (T x)| : y * ∈ Y * , x ∈ X, y * = x = 1, Re y * (Gx) > 1 − δ for every T ∈ L(X, Y). Equivalently, n G (X, Y) is the greatest constant k 0 such that max |w|=1 G + wT 1 + k T for all T ∈ L(X, Y). Here, we first provide some tools to study the numerical index with respect to G. Next, we present some results on the set N (L(X, Y)) of the values of the numerical indices with respect to all norm-one operators on L(X, Y). For instance, we show that N (L(X, Y)) = {0} when X or Y is a real Hilbert space of dimension greater than one and also when X or Y is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. In the real case, we show that for 1 < p < ∞, N (L(X, p)) ⊆ [0, Mp] and N (L(p, Y)) ⊆ [0, Mp] for all real Banach spaces X and Y , where Mp = sup t∈[0,1] |t p−1 −t| 1+t p. For complex Hilbert spaces H 1 , H 2 of dimension greater than one, we show that N (L(H 1 , H 2)) ⊆ {0, 1/2} and the value 1/2 is taken if and only if H 1 and H 2 are isometrically isomorphic. Besides, N (L(X, H)) ⊆ [0, 1/2] and N (L(H, Y)) ⊆ [0, 1/2] when H is a complex infinite-dimensional Hilbert space and X and Y are arbitrary complex Banach spaces. We also show that N (L(L 1 (µ 1), L 1 (µ 2))) ⊆ {0, 1} and N (L(L∞(µ 1), L∞(µ 2))) ⊆ {0, 1} for arbitrary σ-finite measures µ 1 and µ 2 , in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps from a Banach space to itself can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, as constructing diagonal operators between c 0-, 1-, or ∞-sums of Banach spaces, composition operators on some vector-valued function spaces, taking the adjoint to an operator, and composition of operators.

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

Let X be a Banach space and a positive measure. In this article, we show that nðL p ð, XÞÞ ¼ lim m nðl m p ðXÞÞ, 1 p < 1. Also, we investigate the positivity of the numerical index of l p -spaces.