Smooth and path connected Banach submanifold Σ
r
of B(E,F) and a dimension formula in B(?
n
,?
m
) |
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Authors: | Jipu Ma |
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Institution: | (1) Harbin Normal University, Harbin, China;(2) Nanjing University, Nanjing, China;(3) Tseng Yaun Rong Functional Research Center, Harbin Normal University, Harbin, 150080, P. R. China;(4) Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China |
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Abstract: | Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ
r
the set of all operators of finite rank r in B(E,F), and Σ
r
# the number of path connected components of Σ
r
. It is known that Σ
r
is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ
r
. In this paper,the equality Σ
r
# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ
r
is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T
A
Σ
r
= {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ
r
if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property
of Σ
r
the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝ
n
and F = ℝ
m
, then Σ
r
is a smooth and path connected submanifold of B(ℝ
n
, ℝ
m
) and its dimension is dimΣ
r
= (m+n)r−r
2 for each r, 0 <- r < min {n,m}.
Supported by the National Science Foundation of China (Grant No.10671049 and 10771101). |
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Keywords: | operator of finite rank smooth Banach submanifold path connectivity perturbation analysis of generalized inverse |
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