quasieigenvalues

Quasieigenvalues are a concept arising in the study of linear operators, particularly in functional analysis and numerical analysis. They are related to eigenvalues but differ in their definition and application, especially when dealing with non-self-adjoint operators or systems where exact eigenvalues are difficult or impossible to compute.

In simple terms, for a linear operator $A$, an eigenvalue $\lambda$ and its corresponding eigenvector $v$ satisfy the equation $Av = \lambda v$. Quasieigenvalues, however, are often defined in a context where this exact relationship might not hold perfectly. One common interpretation is that they are approximate eigenvalues. In numerical methods, for instance, when approximating the solution to an eigenvalue problem, the computed values might be referred to as quasieigenvalues.

Another perspective on quasieigenvalues emerges when considering operators that are not necessarily bounded or defined on a Hilbert space. In such cases, the standard spectral theory might not apply directly, and quasieigenvalues can represent values that behave similarly to eigenvalues in certain limiting processes or approximations. They can be crucial for understanding the stability and behavior of solutions to differential equations or other dynamical systems. The precise definition can vary depending on the specific mathematical framework being used.