The question you ask is a very broad one. Since differential operators are unbounded - the definition of the on which one defines the operator is important. When one goes to infinite dimensions, there might not even be a point spectrum (eigenvalues) - the spectrum may be only the continuous spectrum and residual spectrum. If the operator has a self joint extension or a normal extension then the problem becomes more tractable but for non-normal operators the problem is quite complex.

At one time a class of operators "spectral operators" were defined, see Dunford and Schwartz, "Linear Operators", vol III, to address the issues on non-normal operators. Proving a non-normal operator - even an ordinary differential operator - was not a trivial task. The goal of this effort was to expand the concept of Jordan Canonical form to first bounded linear operators on a Hilbert/Banach space and then to unbounded linear operators, e.g., differential operators. For compact operators or operators with compact resolvents - that has been solved. For bounded normal and unitary operators, that has been solved (spectral theorem). For unbounded operators with a compact resolvent, that has been pretty much solved (by using the spectral theorem the spectral theorem). For unbounded self adjoint operators, there is also a spectral theorem. However, for non-normal bounded operators and non-self adjoint unbounded operators - there is no general result on a canonical form.

In these problems stability questions be quite problem specific.

This is maybe the best you can do.If you look at the corresponding problem in matrix algebra it corresponds to using the singular value decomposition pseudo

spectrum

1. Stability of the method of lines - University of Oxford

https://people.maths.ox.ac.uk/trefethen/publication/PDF/1992_50.pdf

2. NUMERICAL STABILITY

pub.math.leidenuniv.nl/~spijkermn/NumStab98.pdf

3. INTEGRAL OPERATORS AND DELAY DIFFERENTIAL EQUATIONS 1 ...

https://math.nist.gov/~DGilsinn/publications/Gilsinn_Potra_Int_Eq_Appl.pdf

The question you ask is a very broad one. Since differential operators are unbounded - the definition of the on which one defines the operator is important. When one goes to infinite dimensions, there might not even be a point spectrum (eigenvalues) - the spectrum may be only the continuous spectrum and residual spectrum. If the operator has a self joint extension or a normal extension then the problem becomes more tractable but for non-normal operators the problem is quite complex.

At one time a class of operators "spectral operators" were defined, see Dunford and Schwartz, "Linear Operators", vol III, to address the issues on non-normal operators. Proving a non-normal operator - even an ordinary differential operator - was not a trivial task. The goal of this effort was to expand the concept of Jordan Canonical form to first bounded linear operators on a Hilbert/Banach space and then to unbounded linear operators, e.g., differential operators. For compact operators or operators with compact resolvents - that has been solved. For bounded normal and unitary operators, that has been solved (spectral theorem). For unbounded operators with a compact resolvent, that has been pretty much solved (by using the spectral theorem the spectral theorem). For unbounded self adjoint operators, there is also a spectral theorem. However, for non-normal bounded operators and non-self adjoint unbounded operators - there is no general result on a canonical form.

In these problems stability questions be quite problem specific.

Dear Dr. Dušan Regodić

the references you have introduced talked about the numerical stability.

is there any relation between numerical stability and theoretical stability (by theoretical stability I mean convergence to a steady state solution)?

does numerical stability implies theoretical stability?

Dear Dr. Truman Prevatt

Thank you very much for your comprehensive answer,

you give a good categorization for different operators where their stability can be analyzed by spectral theorem. the operator that I need to analyze is the Fokker-Planck operator in Fokker-Planck equation (Forward-Kolmogorov equation), which in general is a linear parabolic operator with variable coefficients, since you mentioned that stability is problem specific in general, could you please tell me how can I analyze the stability of FPE in general?

There are a lot of concepts of "stability." From your initial question I assumed you were interested in the asymptotic properties and the the case of the parabolic equation FPE the solutions converging as t-->infinity. In such a case there has been a lot of work, In general the FPE is given by df/dt=Af where A is an elliptic operator. If A has a compact resolvent, the the asymptotic properties are determined by the eigenvalues of A. One place you might want to look is Michael Taylor's book on Qualitative properties of PDE's. He has quite a bit on stochastic DE's, FPE, etc.

https://www.amazon.com/Partial-Differential-Equations-Qualitative-Mathematical/dp/1441970517

In general solutions and their behavior of a diffusion equation is determined by the elliptic operator defining the equation. There is a lot of literature on the qualitative and asymptotic behavior of diffusion equations in general and the FPE in specific. Hopefully someone that knows the more recent literature will chime in. I did do a quick search and found the following paper. This may or may not be of any help but there are quite a few references so it might be a place to start.

http://mate.unipv.it/toscani/publi/fokkerplanck.pdf