How does this statement follow? (adjoints on Hilbert spaces)

In summary, adjoints on Hilbert spaces are operators that satisfy a specific property when applied to the space's inner product. They are crucial in functional analysis and quantum mechanics, allowing for the definition of self-adjoint operators with important physical interpretations. The adjoint of a linear operator can be calculated for finite-dimensional Hilbert spaces, but for infinite-dimensional spaces, it requires a deeper understanding of functional analysis. Adjoints on Hilbert spaces are related to Hermitian operators, which are self-adjoint operators. They are unique, but there can be multiple operators with the same adjoint, called adjointable operators.
  • #1
pellman
684
5
If A is an operator on a Hilbert space H and A* is its adjoint, then
upload_2015-8-8_5-53-38.png
. That is, the orthogonal complement of the range of A is the same subspace as the kernel of its adjoint.

Then the author I am reading says it follows that the statements "The range of A is a dense subspace of H" and "A* is injective on Dom(A*)" are equivalent. Can someone explain please?

The operators A and A* are not assumed to be bounded and so their domains may not be all of H and their domains may not be equal to each other.

It could also be that I am misreading this entirely.
 
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  • #3


Based on the given information, it seems that the author is saying that the statements "The range of A is a dense subspace of H" and "A* is injective on Dom(A*)" are equivalent, meaning that they are essentially saying the same thing.

To understand why this is the case, let's break down the statements separately.

First, "The range of A is a dense subspace of H" means that every element in H can be approximated by elements in the range of A. In other words, the range of A is "close" to being all of H.

On the other hand, "A* is injective on Dom(A*)" means that A* is a one-to-one mapping on its domain, meaning that for every element in the domain of A*, there is only one corresponding element in the range of A*. This also implies that the kernel of A* (the set of all elements in the domain that map to 0) is trivial, meaning it only contains the zero vector.

Now, going back to the original statement, if the orthogonal complement of the range of A is the same subspace as the kernel of its adjoint, then it means that the range of A is "close" to being all of H, and the kernel of A* is trivial. This is essentially saying the same thing as the two statements mentioned earlier, hence they are equivalent.

In summary, the given statement is saying that if the range of A is "close" to being all of H and the kernel of A* is trivial, then A* is a one-to-one mapping on its domain. And vice versa, if A* is a one-to-one mapping on its domain, then the range of A is "close" to being all of H and the kernel of A* is trivial.
 

Related to How does this statement follow? (adjoints on Hilbert spaces)

1. How do adjoints on Hilbert spaces work?

The adjoint of an operator on a Hilbert space is the operator that satisfies a certain property when applied to the space's inner product. It is a generalization of the concept of transpose in finite-dimensional vector spaces.

2. What is the importance of adjoints on Hilbert spaces?

Adjoints are crucial in the study of functional analysis and quantum mechanics. They allow for the definition of self-adjoint operators, which have important physical interpretations and applications in mathematical models.

3. Can adjoints on Hilbert spaces be calculated?

Yes, for finite-dimensional Hilbert spaces, the adjoint of a linear operator can be calculated by taking the conjugate transpose of its matrix representation. For infinite-dimensional Hilbert spaces, the calculation is more involved and requires a deeper understanding of functional analysis.

4. How do adjoints on Hilbert spaces relate to Hermitian operators?

Hermitian operators are self-adjoint operators on a Hilbert space. This means that their adjoint is equal to the operator itself, making them particularly important in quantum mechanics where they correspond to observable physical quantities.

5. Are adjoints on Hilbert spaces unique?

Yes, the adjoint of an operator on a Hilbert space is unique. This follows from the uniqueness of the inner product on the space. However, there can be multiple operators that have the same adjoint, known as adjointable operators.

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