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It is possible to slightly modify a compact operator such that it is no longer compact? In particular, I'm looking for explicit expressions for a compact operator and a non-compact operator that are very similar, but due to some minor difference one of them is compact but the other one isn't?

Edit: I am not sure of what notion of similarity would apply in this context. Along with an example I was hoping somebody could suggest what a suitable concept of similarity in this case?

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    Do you measure similarity with the operator norm? – 2017-01-29
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    @LeBtz likely not, as $0+\epsilon\Bbb1$ is not compact. I wouldn't know what kind of a criterium of similarity would be natural here. – 2017-01-29
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    @s.harp I know, that's why I asked. It is crucial information missing in the question, what "slightly modify" means. – 2017-01-29
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    @LeBtz I'm not sure how to measure similarity in such a context. I was hoping that someone would suggest a notion of similarity and provide an example of applying it. – 2017-02-08
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    I've edited my question to say that this vague notion of similarity that I originally mentioned is part of the information I'm looking for..someone to give a proper notion of similarity. Can this question be reopened? – 2017-02-08

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