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

Markus Sigg <mail@xxxxxxxxxx> writes:

> Hi,
>
> let H,U be complex Hilbert spaces, T:H->H an injective positive
> linear operator and F:H->U a compact linear operator. For r>0
> define the power T^r by functional calculus.
>
> Then T^r converges pointwise to the identity for r->0. In general
> we do not have convergence of T^r w.r.t. the operator norm. But do
> we have convergence of FT^r to F w.r.t. the operator norm?

I'd say it's true. It is true for finite--dimensional
operators F, and since we have Hilbert spaces, any compact
operator can be approximated by finite--rank operators
in the operator norm.

> More general: For not necessarily injective T let P be the orthogonal
> projection to the orthogonal complement of the kernel of T. Then T^r
> converges to P pointwise. Does FT^r converge to FP w.r.t. the operator
> norm?

Same argument as above should apply.

> More general: Let F be a Schatten-p operator. Does FT^r converge fo FP
> w.r.t. the Schatten-p norm?

No idea.

Best,
Jakob
.

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