A common property of R(E, F) and B(ℝ n , ℝ m ) and a new method for seeking a path to connect two operators

Abstract

Given two Banach spaces E, F, let B(E, F) be the set of all bounded linear operators from E into F, and R(E, F) the set of all operators in B(E, F) with finite rank. It is well-known that B(ℝn) is a Banach space as well as an algebra, while B(ℝn, ℝm) for m ≠ n, is a Banach space but not an algebra; meanwhile, it is clear that R(E, F) is neither a Banach space nor an algebra. However, in this paper, it is proved that all of them have a common property in geometry and topology, i.e., they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces). Let Σr be the set of all operators of finite rank r in B(E, F) (or B(ℝn, ℝm)). In fact, we have that 1) suppose Σr ∈ B(ℝn, ℝm), and then Σr is a smooth and path-connected submanifold of B(ℝn, ℝm) and dimΣr = (n + m)r − r2, for each r ∈ [0, min{n,m}; if m ≠ n, the same conclusion for Σr and its dimension is valid for each r ∈ [0, min{n, m}]; 2) suppose Σr ∈ B(E, F), and dimF = ∞, and then Σr is a smooth and path-connected submanifold of B(E, F) with the tangent space TAΣr = {B ∈ B(E, F): BN(A) ⊂ R(A)} at each A ∈ Σr for 0 ⩽ r ⩽ ∞. The routine methods for seeking a path to connect two operators can hardly apply here. A new method and some fundamental theorems are introduced in this paper, which is development of elementary transformation of matrices in B(ℝn), and more adapted and simple than the elementary transformation method. In addition to tensor analysis and application of Thom’s famous result for transversility, these will benefit the study of infinite geometry.