Interpolation of operators for $\Lambda$ 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.

Bulletin of the American Mathematical Society

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 K(ct, Ta; B 0 , B 1) [ M 0 ||a|| A0 +e 0 K(t, Ta; B 0 , B 1) and K(ct, Ta; B 1 , B 0) [ M 1 ||a|| A1 +e 1 K(t, Ta; B 1 , B 0). We prove that similar to the classical case for c > 1 and 0 [ e j [ 1 we get for all 0 < h < 1 and for all 0 < q [. ||Ta|| (B0, B1)h, q [ C(M 0 , M 1 , h, q) ||a|| (A0, A1)h, q. If we take B 0 =L 1 (Q), B 1 =L. (Q), c=2, and e j =1, where Q is a cube in R n , we get a theorem of Bennett, DeVore, and Sharpley. We prove that if e j > 1 we continue to get an interpolation theorem, but the interpolation holds for (log + e 0)/(log c) < h < 1 − (log + e 1)/(log c). This is the first instance of an interpolation theorem which holds for a subinterval of 0 < h < 1. Bennett, DeVore and Sharpley identified a ''weak L. '' class as the rearrangement invariant span of BMO(Q). This prompts the natural question of the existence of abstract ''weak type'' classes near the endpoints of interpolation scales. As we note below, there have been previous attempts to develop the theory of such classes. The construction was, however, too rigid, and necessitated the precise identification of the K-functional for the interpolation couple. We define these classes here in a way which allows for the identification K-functionals up to multiplicative equivalence. This opens the door to applications of the theory to most interpolation couples, leading to stronger interpolation theorems even for some well known spaces.

Journal of Approximation Theory, 1969

Journal of Mathematical Analysis and Applications, 1991

Beauzamy in [l] proved that the Lions-Peetre spaces A,, (0 < 0 < 1, 1 <p < co) are reflexive if and only if the imbedding i: d(A) + E(A) is weakly compact. Later on, the Davis-Figiel-Johnson-Pelczynski factorization technique was again used in [lo] to prove that the operator T: A,, + Be,p is weakly compact if and only if T: d(A)-+ Z(g) is weakly compact, thereby generalizing the aforesaid result of Beauzamy. In this paper, a modification of this technique is used to give a characterization of the reflexivity for the more general spaces AH,t.

Studia Mathematica, 2005

We prove an interpolation theorem for weak-type operators. This is closely related to interpolation between weak-type classes. Weak-type classes at the ends of interpolation scales play a similar role to that played by BMO with respect to the L p interpolation scale. We also clarify the roles of some of the parameters appearing in the definition of the weak-type classes. The interpolation theorem follows from a K-functional inequality for the operators, involving the Calderón operator. The inequality was inspired by a K-J inequality approach developed by Jawerth and Milman. We show that the use of the Calderón operator is necessary. We use a new version of the strong fundamental lemma of interpolation theory that does not require the interpolation couple to be mutually closed. 1. Introduction. Weak-type classes were defined in [9], [10], and [11]. Interpolation theorems between these classes give stronger versions of classical interpolation theorems, and calculations of the K-functionals between these classes give a systematic way of proving rearrangement-function inequalities for classical operators. These classes, when constructed at the natural ends of interpolation scales, turn out to be interesting in their own right, and imply useful generalizations of interpolation theorems. The definition of the weak-type classes, Definition 1.2 below, is quite general, and among other things yields an interpolation theorem, Theorem 1.3, that is valid only for a subinterval of the interpolation scale. In this paper we generalize the interpolation theorem. We define weak-type operators between interpolation pairs of Banach groups, and prove an interpolation theorem for these operators. Moreover we prove an inequality between the K-functionals of T a and of a, which implies the interpolation theorem and is of independent interest. We also show that the interpolation

J Funct Anal, 1972

For two pairs of rearrangement invariant spaces D = [(X, , Yr), (X, , Ya)] we give necessary and sufficient conditions for pairs (X, Y) to be weak intermediate for o, i.e., each operator which is of weak types (Xi, YJ, i = 1, 2, also maps X boundedly to Y. Spaces A,(X) are introduced and are shown to have many of the properties that characterize Lorentz L*'r spaces. Necessary and sufficient conditions in terms of a simple function F(s, t) are given in order that (d,(X), A,(Y)) be weak intermediate for o. Other properties of the function F(s, t) yield sufficient conditions and necessary conditions for interpolation theorems.

Arkiv for Matematik, 1989

This originated the question of whether there are generalizations of Theorem L-P to the case Ao~A~ and BomB 1.

We study the operator equation AX = Y , where the operators X and Y are given and the operator A is required to lie in some von Neumann al- gebra. We derive a necessary and sucient condition for the existence of a solu- tionA. The condition is that there must exist a constant K so that, for all nite collections of operatorsfD1;D2;:::;Dng in the commutant, and all collections of vectorsff1;f2;:::;fng ,w e havek Pn j=1DjYfjkKk Pn j=1DjXfjk : We also study the equality kDY fk = KkDXfk, in connection with solving the equation AX = Y where the operator A is required to lie in some CSL algebra.

Transactions of the American Mathematical Society, 2004

In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ H^\infty functions on the unit disc to problems concerning lifting onto H 2 H^2 of an operator T T that is defined on K = H 2 ⊖ ϕ H 2 \mathcal {K} =H^2\ominus \phi H^2 ( ϕ \phi is an inner function) and commutes with the (compressed) shift S S . In particular, he showed that interpolants (i.e., f ∈ H ∞ f\in H^\infty such that f ( S ) = T f(S)=T ) having norm equal to ‖ T ‖ \|T\| exist, and that in certain cases such an f f is unique and can be expressed as a fraction f = b / a f=b/a with a , b ∈ K a,b\in \mathcal {K} . In this paper, we study interpolants that are such fractions of K \mathcal {K} functions and are bounded in norm by 1 1 (assuming that ‖ T ‖ &gt; 1 \|T\|&gt;1 , in which case they always exist). We parameterize the collection of all such pairs ( a , b ) ∈ K × K (a,b)\in \mathcal {K}\times \mathcal {K} and show that each interpolant of this type can be determined as the unique minimum...

Commentationes Mathematicae Universitatis Carolinae, 2016

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Proceedings of the American Mathematical Society, 1988

We apply the complex method of interpolation to families of infinite-dimensional, separable Hubert spaces, and obtain a detailed description of the structure of the interpolation spaces, by means of a unique "extremal" operator-valued, analytic function. We use these interpolation techniques to give a different proof, under weaker assumptions, of a theorem of A. Devinatz concerning the factorization of positive, infinite-rank, operator-valued functions.

Nonlinear Analysis

In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally GΓ-spaces. As a direct consequence of our results any Lorentz-Zygmund space L a,r (Log L) β , is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that 1 < a < ∞, β = 0. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm. Contents 1 Introduction. Main results 2 2 Notations. Preliminary Lemmas 7 3 Computation of some K-functionals and characterization of the interpolation spaces (L p),α , L q),α ) θ,r 11 4 Small Lebesgue space as interpolation of usual Lebesgue spaces 22 5 Interpolation between small, Grand Lebesgue spaces and the associated K-functional 24 6 The critical case p = q. The interpolation space (L p) , L (p ) and its Kfunctional 28

In this paper we are interested in polynomial interpolation of irregular functions namely those elements of L 2 (R, µ) for µ a given probability measure. This is of course doesn't make any sense unless for L 2 functions that, at least, admit a continuous version. To characterize those functions we have, first, constructed, in an abstract fashion , a chain of Sobolev like subspaces of a given Hilbert space H 0. Then we have proved that the chain of Sobolev like subspaces controls the existence of a continuous version for L 2 functions and gives a pointwise polynomial approximation with a quite accurate error estimation.

Annali di Matematica Pura ed Applicata, 1983

In this paper we study those Banach pairs (A0, A1) for which all interpolation is described by Peetre's K- method of interpolation. Special emphasis is given to duality and to the case when (A0, A1) is a pair of K-spaces