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I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?

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    1. Assuming $H$ is separable, it's the unique proper *closed* ideal. (the operators of finite rank are also an ideal, as well as the Schatten $p$-ideals, for instance) 2. An invertible operator is compact if and only if $H$ is finite-dimensional.2011-06-25
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    Since this is a standard result explained in textbooks on operator algebras, is it homework? What have you tried, where are you stuck, what hints did you get, what auxilliary results are you allowed to use?2011-06-25

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