It is shown that there exists a non-separable reflexive Banach space on which every bounded linear operator is the sum of a scalar multiple of the identity operator and an operator of separable range. There is a strong sense that such a Banach space has as few operators as its linear and topological properties allow. IntroductionA long-standing open question is whether there exists a Banach space X on which every operator T : X −→ X is the sum of a scalar multiple of the identity and a compact operator. There are two reasons for asking this question. The first is that in such a Banach space every operator would have pleasant spectral properties. The second reason, and the more important one from the point of view of this article, is that there is a sense that such a Banach space would have 'few' operators.It follows from the axiom of choice, by means of the Hahn-Banach theorem, that any infinite dimensional Banach space is well endowed with linear functionals. As a consequence it is always possible to construct finite rank operators on an infinite dimensional Banach space. If one considers the closure of the space of finite rank operators on a Banach space X, in the operator norm topology, the resulting space is determined entirely by the linear and topological properties of the Banach space X. For instance, if the Banach space X has a Schauder basis, then the closure of the space of finite rank operators in the operator norm topology coincides with the space of compact operators on X. In this case the Banach space X has few operators if every operator on X is of the form λI + K, where λ is a scalar, I is the identity operator and K is a compact operator.A more extreme few-operators question is whether there exists an infinite dimensional Banach space on which every operator is the sum of a scalar multiple of the identity and a nuclear operator.Other few-operator questions may be asked. Is there a non-separable Banach space X in which every operator is the sum of a scalar multiple of the identity and an operator of separable range? Is there a non-reflexive space on which every operator is the sum of a scalar multiple of the identity and a weakly compact operator? Is there a Banach space on which every operator is the sum of a scalar multiple of the identity and a strictly singular operator [3]?Thus it is natural to search for a Banach space X where every operator is a small perturbation of a scalar multiple of the identity, where the notion of smallness will depend on the linear and topological properties of the Banach space X.

It is shown that if X is an infinite-dimensional Banach space with a boundedly complete subsymmetric basis, then the infinite l ∞ direct sum (X ⊕ X ⊕ X ⊕ · · ·) l ∞ is primary.A Banach space X is called primary if for every bounded linear projection P on X either P (X) or (I − P )(X) is isomorphic to X. A Banach space X is called prime if for every bounded linear projection P on X with dim P (X) = ∞, P (X) is isomorphic to X. These are the only prime Banach spaces discovered to date.The Banach spaces L p [0, 1] (1 < p < ∞) [1] and many other classical Banach spaces have been shown to be primary. In this paper we are concerned with Banach spaces having bases with a certain property.A sequence (e n ) ∞ n=1 in a Banach space X is called a basis of X if for every x ∈ X there is a unique sequence of scalars (a n ) ∞ n=1 so that x = ∞ n=1 a n e n . A basis (e n ) ∞ n=1 in a Banach space X is called monotone if, for every choice of scalars (a n ) ∞ n=1 , the sequence of numbers

A subset of a Banach space is called equilateral if the distances between any two of its distinct elements are the same. It is proved that there exist nonseparable Banach spaces (in fact of density continuum) with no infinite equilateral subset. These examples are strictly convex renormings of 𝓁 1 ([0, 1]). A wider class of renormings of 𝓁 1 ([0, 1]) which admit no uncountable equilateral sets is also considered. M S C 2 0 2 0 46B20, 03E75, 46B26 (primary)

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