How does one-to-oneness follow from this?

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In summary, the author is questioning how A - λI being injective follows from the fact that for all . They ask for an explanation and another person responds by explaining that an operator T is injective if its kernel only contains the zero vector. They then use a contradiction argument to show that no non-zero vector exists in the kernel of A - λI. This allows them to use a theorem to prove that A - λI is injective.
  • #1
pellman
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Given a symmetric operator A on a Hilbert space with inner product
upload_2015-8-3_19-47-29.png
and a complex number λ = a + ib, we know that
upload_2015-8-3_19-48-56.png
for all
upload_2015-8-3_19-49-18.png
.

The author I am reading then says: this shows that A - λI is injective ( one to one). I don't see how this follows. Can someone explain?
 

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  • #2
Hello. Recall that an operator ##T## is injective iff ##ker(T)=\{ \textbf{0}\} ## (the set containing only the zero vector). By contradiction, if there were a ##\psi \neq \textbf{0}## such that ##(A-\lambda I)\psi=0##, then, recalling that ##\langle \psi, \psi \rangle >0##, we would have ##\langle (A-\lambda I)\psi , (A-\lambda I)\psi \rangle=0< b^2\langle \psi, \psi \rangle##. This establishes the contradiction, so no such non-zero vector exists in the kernel. Now that we know ##ker(A-\lambda I)=\{\textbf{0} \}## we can use the theorem relating kernel and injectivity of an operator to get that ##A-\lambda I## is injective.
 
  • #3
Thank you very much.
 

FAQ: How does one-to-oneness follow from this?

1. How does one-to-oneness follow from this?

One-to-oneness is a concept that refers to the idea that every element in a set has a unique corresponding element in another set. It follows from this because in order for one-to-oneness to exist, each element in one set must be paired with a unique element in another set, creating a one-to-one correspondence between the elements.

2. What is the significance of one-to-oneness in scientific research?

One-to-oneness is important in scientific research because it allows for accurate and precise measurements and comparisons. By ensuring that each element has a unique counterpart, scientists can confidently make conclusions and predictions based on their data.

3. How does one-to-oneness relate to the scientific method?

The scientific method relies on accurate and objective data collection and analysis. One-to-oneness ensures that each data point is unique and can be used to make reliable conclusions and predictions.

4. Can one-to-oneness be observed in nature?

Yes, one-to-oneness can be observed in various natural phenomena. For example, in genetics, each gene has a specific function and corresponds to a unique trait, creating a one-to-one relationship. Additionally, in chemistry, each element has a unique atomic number and properties, leading to a one-to-one correspondence between the elements.

5. How does one-to-oneness impact the understanding of complex systems?

One-to-oneness is crucial in understanding complex systems as it allows for a clear and organized representation of the relationships between different elements. By establishing a one-to-one correspondence, scientists can better analyze and interpret the interconnectedness of various components in a system.

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