We prove an approximate spectral theorem for non-self-adjoint operators and investigate its applications to second order differential operators in the semi-classical limit. This leads to the construction of a twisted FBI transform. We also investigate the connections between pseudospectra and boundary conditions in the semi-classical limit. AMS subject classification numbers: 81Q20, 47Axx, 34Lxx.

The generation of spectral portraits requires the speciication of a region in the complex plane which must be predetermined. For cases where little is known a priori about the spectral properties of a matrix, an accurate and eecient means to compute the outer boundaries of the pseudospectra is desirable. It is shown that the eld of values is an eeective way to determine the extremities of the p...

Eigenvalue analysis plays an important role in understanding physical phenomena. However, if one deals with strongly nonnormal matrices or operators, the eigen-values alone may not tell the full story. A popular tool which can be useful to get more insight in the reliability or sensitivity of eigenvalues is "-pseudospectra. Apart from "-pseudospectra we consider other tools which might help to ...

The concept of "{pseudospectra for matrices, introduced by Trefethen and his co-workers, has been studied extensively since 1990. In this paper, "{ pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as the practical computation of "{pseudospectra for matrix pencils will be discussed. As an application,...

Let g: E → F be an analytic function between two Hilbert spaces E and F. We study the set g(B(x, ε)) ⊂ F, the image under g of the closed ball about x ∈ E with radius ε. When g(x) expresses the solution of an equation depending on x , then the elements of g(B(x, ε)) are ε-pseudosolutions. Our aim is to investigate the size of the set g(B(x, ε)). We derive upper and lower bounds of the following...