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

On Thu, 21 Jul 2005 01:44:35 +0200, Markus Sigg <mail@xxxxxxxxxx>
wrote:

>David C. Ullrich wrote:
>
>> Because the identity is not compact.
>
>Yes, and there exist compact injective T, e.g. T(x_n) := (x_n/n)
>on ell_2. For injective T we have T^r \to identity pointwise.
>But not necessarily in norm, because else the identity would be
>compact.
>
> > (T^r _is_ a norm-continuous
>> function of r for r > 0, right? Just a guess based on the
>> heuristic...)
>
>Because the continuous functional calculus is, well, continuous.

Heh-heh.

>(Is it called "continuous functional calculus" because it is
>continuous, or because it is defined on the space of continuous
>functions defined on the spectrum of T?)

I believe it's the latter, although the former would be more
fun, as above.

>Markus


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

David C. Ullrich
.

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