Abstract: Associated with the construction of intermediate Banach spaces in interpolation theory are unbounded non-linear operators, Ω, which are, roughly, the differentials of the fundamental mappings used in the construction of the intermediate spaces. We look at the theory of these operators and their applications to interpolation theory and to other topics in analysis. We see that the mapping properties of Ω are strongly related to the structure of the interpolation scale. We also see that computations using Ω are closely related to certain computations in the theory of hypercontractive semigroups.

Introduction and Summary: Several constructions in interpolation theory start with a pair of Banach spaces (for example, L1 and L∞) and construct a parameterized family of spaces (in the example, Lp, 1 < p < ∞) which are, in an appropriate technical sense, intermediate to the original pair. In particular it will be true that if a linear operator T is bounded on both spaces of the starting pair, then T will also be bounded on the intermediate spaces.

Associated to the construction of intermediate spaces are mappings Ω, generally unbounded and non-linear, which can be obtained by differentiation with respect to certain parameters used in the construction of the intermediate spaces.

Abstract. We study some questions raised in theory of complex interpolation of quasi-Banach spaces. In particular we give a criterion for the interpolated space to be locally convex.

1. Introduction. In [1] and [2], Coifman, Cwikel, Rochberg, Saghar and Weiss introduced and studied complex interpolation of families of Banach spaces. Recently, Tabacco [11],[12] and Rochberg [10] have studied the extension of these ideas to the non-locally convex quasi-Banach case.

We let T denote the unit circle in the complex plane and λ denote normalizecf Haar measure on T, i.e. dλ = (2π)−1dθ. Δ denotes the unit disk, {z : ∣z∣ < 1}. We then suppose that we are given a family of quasinormed spaces Xw for w ∊ T and define interpolation spaces Xz for z ∊ Δ. The precise details of the construction are given in Section 2.

In this paper we prove two main results on interpolation of analytic families of quasi-Banach spaces. In Theorem 4, we answer a question of Rochberg [10] by giving a condition for the interpolated space to be locally convex. We use here the notion of (Rademacher) type. A quasi-Banach space X is of type p where 0 < p ≤ 2 if there is a constant C so that if x1,…, xn ∊ X then

where the signs ∊k = ±1 are chosen at random. In fact if p < 1 then type p is equivalent to p-normability [5], but there are type one spaces which are not locally convex (e.g. the Lorentz spaces where 1 < p < ∞, or the Ribe space [5]).

Abstract.

An unbounded non-commutative monotone convergence result (Theorem 9) is proved. A new derivation of the Friedrichs extension is given. The basics of the Takesaki cones are studied.

1. Introduction.

We study certain aspects of the theory of von Neumann algebras that emphasize its interpretation as non-commutative measure theory. In this interpretation, which is direct and unmistakable, the projections in the algebra are the characteristic functions of the (non-commuting) measurable sets (which do not appear!), the elements of R are the bounded measurable functions, and the (normal) states of R are the probability measures on the underlying (non-commutative) measure space (which, again, does not appear). An important result, in the early stages of the theory, states that if {An} is a monotone increasing sequence of self-adjoint operators, bounded above (by some multiple of the identity operator I), then there is a bounded self-adjoint operator A such that Anx → Ax for each x in the underlying Hilbert space. If each An ∊ R then, of course, A ∊ R (cf. [6; Lemma 5.1.4]). This is a primitive non-commutative monotone convergence theorem. In Section 3, the restriction that the sequence {An} be bounded above is removed; the limit A is now an appropriate unbounded self-adjoint operator affiliated with R. (We write A η R to indicate this affiliation.) The extension of the classical bounded non-commutative monotone convergence result to this unbounded version (Theorem 9) seems not to be routine.

The Special Year in Modern Analysis at the University of Illinois was devoted to the synthesis and expansion of modern and classical analysis. The program brought together analysts from around the globe for intensive lectures and discussions, including an International Conference on Modern Analysis, held March 16–19, 1987. The Special Year's success is a tribute to the outstanding merits and professional dedication of the participants. Contributions to these Proceedings of the Special Year were solicited from the participants in order to record and disseminate the fruits of their activities. The editors are grateful to the contributors for their response, which accurately reflects the quality and substance of the Special Year. In keeping with the wide scope of topics treated, the contents of these Proceedings fell naturally into two interrelated volumes, covering “Analysis in Abstract Spaces” and “Analysis in Function Spaces”.

Thanks are due to the National Science Foundation, the Argonne Universities Association Trust Fund, the University of Illinois Campus Research Board, the University of Illinois Miller Endowment Fund, the University of Illinois Department of Mathematics, and J. Bourgain's Chair in Mathematics at the University of Illinois, without whose financial support the Special Year could not have taken place. Special thanks are also due to Professor Bela Bollobas, Consulting Editor at Cambridge University Press, and Mr. David Tranah, Senior Editor in Mathematical Sciences at Cambridge University Press, for the guidance and encouragement which made these Proceedings possible.

Every linear map on Cn has an upper triangular form; and for a fixed basis, the set of upper triangular matrices is a tractable object. For operators on Hilbert space, the notion of triangular form is replaced by the search for a maximal chain of invariant subspaces. This has been a rather intensive search, but the Invariant Subspace Problem remains, and is likely to remain for some time. The study of nest algebras takes the other point of view: fix a complete chain of closed subspaces (a nest) and study the algebra of all operators leaving each element of the nest invariant. That is, we study all operators with a given triangular form.

This sub-discipline of operator theory is about twenty five years old. It has reached a stage where there are many nice results, and a fairly satisfactory theory. Yet there are still interesting and compelling problems remaining. In these lectures, I will attempt to describe some of the results and to state some of these open questions.

Closely related to nest algebras are the so called CSL algebras. A CSL is a complete lattice L of commuting projections. The associated algebra Alg L consists of all operators leaving the ranges of L invariant. That is, all operators A such that P⊥-AP = 0 for all P in L.

Having established the necessary background, we cover a variety of topics in this chapter. Starting from the definition of a Hankel matrix we give three equivalent approaches to the task of defining a Hankel operator on H2 – that is, an operator whose matrix is a Hankel matrix with respect to the usual basis, {1, z, z2, …}. All three approaches have been used in the literature and we choose what is in some ways the simplest, explaining how one can easily pass from this to the others.

The first big theorem is Nehari's theorem, which associates with a bounded Hankel operator a function in L∞(T) (a symbol) whose norm is the same as the operator norm. We give some examples, including Hilbert's Hankel matrix.

Next we come to two famous problems of complex analysis, the Carathéodory-Fejér and Nevanlinna-Pick problems, which we state in their simplest forms (many more difficult versions have been analysed). Each can be reduced to the Nehari extension problem – that of finding a symbol of minimum norm – and we give Sarason's elegant solution to this.

Turning to the more general theory of Hankel operators, we give Kronecker's theorem, characterising finite-rank Hankel operators in terms of rational symbols. We then prove Hartman's theorem characterising compact Hankel operators – a much deeper result. En route we introduce the disc algebra, a subspace of H∞.

Much of the material of this chapter is covered by the cited works of Francis, Garnett, Power and Sarason.