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

In summary, for a power of an operator to exist, the domain of the operator must remain invariant under the operation and the range must be included in the domain to allow for consecutive application. This is important in defining the generalized eigenspace of an unbounded operator and can be formalized through the Borel Functional Calculus.
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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
 
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fxdung said:
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.
 
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  • #4
fresh_42 said:
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.
 
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Look up the Borel Functional Calculus. It formalizes the general idea of applying general Mathematical operations to linear operators.
 
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1. What is a linear operator?

A linear operator is a mathematical function that maps one vector space to another, while preserving the basic structure of the vector space.

2. What is meant by "raising a linear operator to some power"?

Raising a linear operator to some power means applying the operator to itself a certain number of times, as determined by the power. For example, raising a linear operator A to the second power would mean applying A to itself twice: A*A.

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

In order to raise a linear operator to some power, the operator must be defined on a vector space and the power must be a positive integer. Additionally, the vector space must have a defined multiplication operation.

4. Can any linear operator be raised to any power?

No, not all linear operators can be raised to any power. The power must be a positive integer and the operator must be defined on a vector space with a defined multiplication operation. Additionally, some linear operators may have restrictions or limitations that prevent them from being raised to certain powers.

5. What are some applications of raising a linear operator to some power?

Raising a linear operator to a power can be useful in solving systems of linear equations, finding eigenvalues and eigenvectors, and in applications such as signal processing and image processing. It can also be used to represent repeated transformations in linear algebra and functional analysis.

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