I What conditions are needed to raise a linear operator to some power?

[fxdung]

Each operator has a domain, so for a power of an operator to exist, the domain of the operator must remain invariant under the operation.

Is that correct?

mentor note: edited for future clarity

 

[jedishrfu]

This may answer your question:

https://math.stackexchange.com/questions/453914/linear-map-vs-operator-raised-to-power

 

[fresh_42]

Each operator has a domain, so for a power of an operator to exist, the domain of the operator must remain invariant under the operation.

Is that correct?

mentor note: edited for future clarity

The range must be included in the domain, simply to allow a consecutive application.

 

[S.G. Janssens]

The range must be included in the domain, simply to allow a consecutive application.

This is the wrong approach if $D(A)$ is a proper subspace of a vector space $X$.

For example, take $X = C[0,1]$ and $D(A) = C^1[0,1]$ and $A : D(A) \to X$ differentiation. Then the range of $A$ is not contained in its domain, but $A^2$ is well-defined with
$$
D(A^2) = C^2[0,1] = \{f \in D(A)\,:\,Af \in D(A)\}.
$$
(This also suggests the general definition.) This is important if one wants to talk about the generalized eigenspace of an unbounded operator, which is defined in terms of positive integer powers of $\lambda I - A$, where $I$ is the identity on $X$ and $\lambda$ is an eigenvalue.

 

[WWGD]

Look up the Borel Functional Calculus. It formalizes the general idea of applying general Mathematical operations to linear operators.