Re: compact operators, convergence pointwise / w.r.t. operator norm

On Wed, 20 Jul 2005 11:35:00 +0200, Markus Sigg <mail@xxxxxxxxxx>
wrote:

>Jakob Creutzig wrote:
>
>>>More general: Let F be a Schatten-p operator. Does FT^r converge fo FP
>>>w.r.t. the Schatten-p norm?
>>
>>
>> No idea.
>
>It can be reduced to the operator norm case, again using finite rank
>approximations, and looking at the singular values.
>
>BTW, the argument "it is true for finite-rank operators, and we can
>approximate any compact operator by finite rank operators, so it is
>true for compact operators" has to be used with care. Even for compact
>positive T, T^r does not necessarily converge in operator norm for
>r \to 0.

Because the identity is not compact. (T^r _is_ a norm-continuous
function of r for r > 0, right? Just a guess based on the
heuristic...)

>Markus


************************

David C. Ullrich
.