Abstract. A problem of enduring interest in connection with the study of frames in Hubert space is that of characterizing those frames which can essentially be regarded as Riesz bases for computational purposes or which have certain desirable properties of Riesz bases. In this paper we study several aspects of this problem using the notion of a pre-frame operator and a model theory for frames derived from this notion. In particular, we show that the deletion of a finite set of vectors from a frame {jc,,}^, leaves a Riesz basis if and only if the frame is Besselian (i.e., SSii anXn converges <=> (a") e I2).

Abstract.Let E and F be Banach spaces and denote by L(E, F) (resp., K(E, F)) the space of all bounded linear operators (resp., all compact operators) from E to F. In this note the following theorem is proved: If E and F are reflexive and one of E and F has the approximation property then the following are equivalent:This result extends a recent result of Ruckle (Proc. Amer. Math. Soc. 34 (1972), 171-174) who showed (i) and (ii) are equivalent when both E and F have the approximation property. Moreover the proof suggests strongly that the assumption of the approximation property may be dropped.The purpose of this note is to call attention to an unsolved problem in Banach space theory whose complete solution seems to be quite elusive and to make a contribution toward the complete solution by opening a new avenue of approach.Let E and F be Banach spaces and denote by L(E, F) (resp., K(E, F)) the space of all bounded linear operators (resp., all compact operators) from E to F. A problem which is as yet unsolved is: Characterize those spaces L(E, F) which are reflexive. A partial solution has been given in [4] and [7], namely: Theorem 1. If E and F are reflexive and both E and F have the approximation property then L(E, F) is reflexive if and only if L(E, F) = K(E, F).Our purpose here is to prove a result (Theorem 2) which is both an extension of, and an improvement on, Theorem 1. In particular, we give another characterization of those spaces L(E, F) which are reflexive and at the same time show that Theorem 1 is valid under the weaker assumption that either E or Phas the approximation property. Also, our proof avoids

Let E be a normed linear space and suppose x ~ E with Ilx[[ = 1. Then x is called an extreme point of the unit ball in E if it is not the midpoint of a line segment lying in the unit ball of E, and is called a smooth point if there exists a unique hyperplane in E tangent to the unit ball at x.In less geometric language, x is an extreme point if whenever x = i/2x 1 -I-l/2x 2 with ]Ix 1 II = llx2l[ = 1 then x 1 = x2, while x is a smooth point if there exists a unique f~ E* such that tlfll = 1 and ( f , x) = Ilxll = 1. The study of extreme points and smooth points on the unit ball of a normed space has attracted considerable attention for many years due to the fact these concepts have been shown to be important tools in the investigation of the metric structure of normed spaces (for an excellent survey of results along these lines see [1] or [4]).In a paper published in 1957 [7] Schatten proved the interesting result that the unit ball in the space K(: 2, :2) of all compact operators on :2 has no extreme points (and hence by the Krein-Milman ttieorem cannot be a conjugate space). The purpose of this note is to continue the study of the geometry of operator spaces on :2 by characterizing the smooth points on the unit balls in K(: 2, :2) and in its dual space N(: 2, :2) (the space of nuclear, or trace class, operators on :z) and the extreme points on the unit ball in N ( : 2, :2) in terms of properties of the operators themselves. Our basic tools in these investigations will be the so-called "polar representation" of a compact operator on :z (which was used by Schatten in his proof of the above quoted result) and methods of topological tensor products. § 2. Notation and Preliminary ResultsThroughout the paper the term "operator" or "map" will refer to a continuous linear transformation. If T is an operator on :2 we denote its usual norm by I[TH and its nuclear norm by I[T[TN. We will denote the space of all continuous linear operators by ~( : 2 , ~,2), the space of compact operators by K ( : 2, :2), and the space of nuclear (trace class)

A definition of an isometric shift operator on a Banach space is given which extends the usual definition of a shift operator on a separable Hilbert space. It is shown that there is no such shift on many of the common Banach spaces of continuous functions. The associated ideas of a semi-shift and a backward shift are also introduced and studied in the case of continuous function spaces.

called the tensor product of S and T or the tensor product mappin9 [3~ p. 37]. If e denotes the least crossnorm on X@ Y and n denotes the greatest crossnorm, then S ®~ T : X 1 ®~ 1 ~] ~ X 2 ®~ Y2 and S @~ T : X 1 @~ Y1 ~ X2 ®, Y2 are both continuous linear operators [3, p. 93 and 37]. Grothendieck has shown that if S and T are isomorphisms (i.e. linear homeomorphisms) then S @, Tis also an isomorphism, but S @~ T need not be an isomorphism [3, p. 93]. On the other hand, if S and T are topological homomorphisms onto dense subspaces of X2 and ]12 respectively, then S @, T is a topological homorphism which is onto but S@~ T is not necessarily so [3, p. 39].In § 3 we continue the study of tensor product mappings begun in [3] by considering tensor products of p-absolutely summing [6], nuclear [8] and quasi-nuclear I-9] maps. In a recent paper [6] Lindenstrauss and Pelczynski have demonstrated the power of the notion of p-absolutely summing mappings in investigating the subspace structure of Banach spaces, while Grothendieck 1-3] and Pietsch 1-8], [9], have extensively studied nuclear and quasi-nuclear maps.The main results of § 3 are (i) S ®~ T is p-absolutely summin9 if and only ifS and T are p-absolutely summing (1 < p < + ~). However an example is given of absolutely summing maps S and T for which S@,T is not p-absolutely summing for any 1 < p < + ~. (ii) If e is any crossnorm (~ ~ e < ~) then S @~ T is nuclear if and only if S and T are nuclear. (iii) S®, T is quasi-nuclear if and only if S and T are quasi-nuclear.Again, an example is given to show that S and T may both be quasinuclear and yet S @~ T need not be quasi-nuclear.Examples are given of classes of operators A for which S and T are in A but S@~ T is not in A, providing a contrast to the results (i), (ii), and (iii) quoted above.In § 4 we use the notion of tensor product mapping to show that certain results of Schatten [12] concerning subspaces of X®~ Y also hold for X®~ Y where e is any uniform crossnorm.