Smooth and path connected Banach submanifold Σ r of B(E,F) and a dimension formula in B(ℝ n ,ℝ m )

Abstract

Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σr the set of all operators of finite rank r in B(E,F), and Σ#r the number of path connected components of Σr. It is known that Σr is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σr. In this paper,the equality Σ#r = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σr is a smooth and path connected Banach submanifold in B(E,F) with the tangent space TAΣr = {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σr if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of Σr the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝn and F = ℝm, then Σr is a smooth and path connected submanifold of B(ℝn, ℝm) and its dimension is dimΣr = (m+n)r−r2 for each r, 0 <- r < min {n,m}.