I What's a Non-Eigenvalue Problem Called?

[member 428835]

Hi PF!

So if this $L[\phi] = M[\phi]$ is called an eigenvalue problem (where $L,M$ are operators, could be differential or matrices) then what is this called $L[\phi] = M$?

 

[fresh_42]

Hi PF!

So if this $L[\phi] = M[\phi]$ is called an eigenvalue problem (where $L,M$ are operators, could be differential or matrices) then what is this called $L[\phi] = M$?

If $(L-M).\Phi = 0$ then $\Phi$ is an eigenvector of $L-M$ to the eigenvalue $0$, i.e. an element in the kernel in case $L,M$ are linear.

Now, what is where from? Since $L\Phi = M$ makes no sense, they aren't in the same vector space anymore!

The general situation is a bit complexer than white and black: see e.g. https://www.physicsforums.com/insights/hilbert-spaces-relatives-part-ii/

 

[member 428835]

If $(L-M).\Phi = 0$ then $\Phi$ is an eigenvector of $L-M$ to the eigenvalue $0$, i.e. an element in the kernel in case $L,M$ are linear.

Now, what is where from? Since $L\Phi = M$ makes no sense, they aren't in the same vector space anymore!

The general situation is a bit complexer than white and black: see e.g. https://www.physicsforums.com/insights/hilbert-spaces-relatives-part-ii/

I guess what I'm wondering is, what is this equation formally called $A x = b$? Clearly it's not an eigenvalue problem.

 

[fresh_42]

I guess what I'm wondering is, what is this equation formally called $A x = b$? Clearly it's not an eigenvalue problem.

Again, depends on what is where from. Usually, i.e. if written this way, it is just an inhomogeneous system of linear equations. This remains the case if the vector space for the coefficients is an ordinary number field or a Hilbert space of functions and $A$ a differential operator.

As you have posed this questions under differential equations, then this imposes restrictions on $x$. A linear differential equation of this type would be a vector $x_i=y^{(i)}$ and the coefficients of $A$ and $b$ e.g. functions $\mathbb{R} \longrightarrow \mathbb{R}$.

 

[HallsofIvy]

An equation of the form "Ax= b" is simply a "linear equation".

 

[Mark44]

In response to the thread title, "What's a non-eigenvalue called?"...

This is akin to asking, "What's a non-zebra called?" There are lots of things that aren't zebras -- we don't have any special names for these things. The same is true for non-eigenvalues.

 

[fresh_42]

So if this $L[\phi] = M[\phi]$ is called an eigenvalue problem (where $L,M$ are operators, could be differential or matrices) then what is this called $L[\phi] = M$?

I don't think there is a name for it. The eigenvalue problem affects all linear functions, and it is a specific equation behind it, a property. Things which don't have this property are normally not named.

There is an exception in functional analysis, where non eigenvalues (together with a condition about boundedness) are called resolvents. But I haven't heard of e.g. the resolvent problem.

 

[S.G. Janssens]

There is an exception in functional analysis, where non eigenvalues (together with a condition about boundedness) are called resolvents. But I haven't heard of e.g. the resolvent problem.

If $A$ is a closed, linear operator on a complex Banach space, then

• Its resolvent set is the set $\rho(A) := \{z \in \mathbb{C}\,:\, (z I - A)^{-1} \text{ exists as a bounded linear operator}\}$. Points in the resolvent set are called regular values.
• For $z \in \rho(A)$ the operator $(z I - A)^{-1}$ is called the resolvent operator and the map $z \mapsto (z I - A)^{-1}$ is called the resolvent map.
• The complementary set $\sigma(A) := \mathbb{C} \setminus \rho(A)$ is called the spectrum of $A$. Of course, in finite dimensions it coincides with the set of eigenvalues, but in general it may contain much more (and possibly even no eigenvalues at all).

So, a "non-eigenvalue" is never called a "resolvent". It may be a regular value, or it may be another point in the spectrum of $A$.

Usually, the "resolvent problem" is just the task of determining the resolvent operator for a particular $A$, but that terminology is indeed less used.

 

[MathematicalPhysicist]

In response to the thread title, "What's a non-eigenvalue called?"...

This is akin to asking, "What's a non-zebra called?" There are lots of things that aren't zebras -- we don't have any special names for these things. The same is true for non-eigenvalues.

Obviously we have a special name for them, they are called "non-zebras".