Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible operators in $B(H)/K(H)$, $Inv(B(H)/K(H))\_0$ --- connected component of $id$ in $Inv(B(H)/K(H))$. $ind\colon Inv(B(H)/K(H))\to \mathbb{Z}$ --- Fredholm index.


I want to find a reference for the following fact:



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> Fact 1. If $H$ is infinite-dimensional and separable then $ind$ is locally-constant and provides an isomorphism between $Inv(B(H)/K(H))/Inv(B(H)/K(H))\_0$ and $\mathbb{Z}$.
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In Murphy's [textbook](http://rads.stackoverflow.com/amzn/click/0125113609), where I've read almost all I know about Fredholm index, there is [Atkinson's theorem](http://en.wikipedia.org/wiki/Atkinson%27s_theorem), the fact, that Fredholm index is locally constant and $ind(ab)=ind(a)+ind(b)$.
But in order to check the fact 1, I need also the fact, stated in the head of the question, or, equivalently, the following:



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> Fact 2. If $H$ is infinite-dimensional and separable then the set $\{a\in B(H)/K(H)\mid ind(a)=0\}$ is connected.
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Where can I find the references? Where can I read that?



Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
You can use some spectral theory to show that the set of unitary operators in $B(H)$ is path-connected. Then the (path-)connectedness of the invertibles follows easily from the polar decomposition. See Theorem 5.29 and Corollary 5.30 in Douglas's *Banach Algebra Techniques* (1st edition).


