Restrictions of compact operators

In summary, if T: X\to Y is a compact linear operator and N is a subspace, then T|_N (the restriction of T to N) is also compact. This holds true because if B is a bounded subset of N, it will also be a bounded subset of X, making its image precompact in Y. However, if N is just an arbitrary subset of X, the restriction may not make sense as a linear operator. In this case, T would still make sense on the closed linear span of N.
  • #1
AxiomOfChoice
533
1
Is it true that if [itex]T: X\to Y[/itex] is a compact linear operator, [itex]X[/itex] and [itex]Y[/itex] are normed spaces, and [itex]N[/itex] is a subspace, then [itex]T|_N[/itex] (the restriction of [itex]T[/itex] to [itex]N[/itex]) is compact? It seems like it would work, since if [itex]B[/itex] is a bounded subset of [itex]N[/itex], it's also a bounded subset of [itex]X[/itex] and hence its image is precompact in [itex]Y[/itex].

But what if [itex]N[/itex] is just an arbitrary subset of [itex]X[/itex]? I guess it doesn't work in that case, though, since it doesn't even make sense to talk about [itex]T[/itex] being a linear operator in that case.
 
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  • #2
Yes,the restriction will be compact by definition. When N is arbitrary, T would still make sense on the closed linear span of N.
 

Related to Restrictions of compact operators

1. What are compact operators?

Compact operators are linear transformations between Banach spaces that map bounded sets to relatively compact sets. They are important in functional analysis and have applications in fields such as differential equations and quantum mechanics.

2. How are compact operators different from other linear operators?

Unlike other linear operators, compact operators have the property that the image of any bounded set is relatively compact. This means that the closure of the image is compact, which is a stronger condition than just being closed. Additionally, compact operators are not necessarily bounded, unlike most linear operators.

3. How do you determine if a given operator is compact?

There are several criteria for determining if an operator is compact, such as the Arzelà–Ascoli theorem and the Riesz lemma. In general, an operator is compact if it maps bounded sequences to sequences with convergent subsequences. However, there are also specific conditions for different types of operators, such as compactness being equivalent to being Hilbert-Schmidt for self-adjoint operators.

4. What are some applications of compact operators?

Compact operators have various applications in mathematics and physics. In functional analysis, they are used to prove theorems about Banach spaces, such as the Riesz representation theorem. In differential equations, they can be used to study the behavior of solutions and in numerical analysis, they are important for approximating solutions. In quantum mechanics, they are used to describe the evolution of quantum systems.

5. Can compact operators be represented by matrices?

Yes, in finite-dimensional spaces, compact operators can be represented by matrices. However, in infinite-dimensional spaces, compact operators cannot always be represented by matrices. This is because the image of a compact operator must be infinite-dimensional, which cannot be captured by a finite-dimensional matrix. In general, compact operators are not limited to finite-dimensional spaces and have a much broader range of applications in infinite-dimensional settings.

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