Eigenvalues of operator between L^2

In summary, the problem is asking for the eigenvalues and eigenvectors of the linear transformation Mf(t) = ∫(-π,π) sin(y-x)f(x) dx, and how to use this information to determine the spectrum. The student has attempted to construct a particular form of f(x) using trigonometric functions, but has encountered a problem with integrating and is unsure if this is the correct approach. They are seeking clarification on the steps to take to find the desired information.
  • #1
Raven2816
20
0

Homework Statement



>M: L_2 -> L_2
>
>(Mf)(t) = int(-pi, pi) sin(y-x)f(x) dx
>
>how do i find eigenvalues/vectors of M and what can i use to find
>information about the spectrum?



Homework Equations





The Attempt at a Solution



now i know that sin(y-x) = sinycosx-cosysinx
i also realize that the range is 2dimensional
when i went to construct f i made it = cosy + asinx
so i plugged this in, but when i integrated i got 0. did i integrate wrong? or did i take the wrong approach?
 
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  • #2
Can you give the steps that led you to the particular form of f(x) that you have stated?
 
  • #3
because i know that the eigenvector is contained in a subspace spanned by sin(x) and cos(x). i figured that'd be a good f?
 

FAQ: Eigenvalues of operator between L^2

1. What are eigenvalues of an operator between L^2?

The eigenvalues of an operator between L^2 are the set of values that, when multiplied by the operator, produce the original function multiplied by a constant. In other words, they are the values for which the operator acts as a scaling factor on the function.

2. How are eigenvalues of an operator between L^2 calculated?

Eigenvalues of an operator between L^2 can be calculated by finding the roots of the characteristic equation, which is obtained by setting the determinant of the operator's matrix representation equal to zero.

3. What is the significance of eigenvalues of an operator between L^2?

The eigenvalues of an operator between L^2 have a number of important applications in mathematics and physics. They are used to solve differential equations, study the stability of dynamical systems, and understand the behavior of quantum mechanical systems.

4. How do eigenvalues of an operator between L^2 relate to eigenvectors?

Eigenvalues and eigenvectors are closely related. The eigenvectors of an operator between L^2 are the functions that correspond to each eigenvalue. In other words, they are the functions that, when multiplied by the operator, result in the eigenvalue times the original function.

5. Can an operator between L^2 have an infinite number of eigenvalues?

Yes, an operator between L^2 can have an infinite number of eigenvalues. This is because L^2 is an infinite-dimensional vector space, and there can be an infinite number of distinct eigenvalues for any given operator in an infinite-dimensional space.

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