Spectral Properties of Non-Self-Adjoint Perturbations for a Spectral Problem with Involution

Spectral Properties of Random Non-self-adjoint Matrices and Operators

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the ...

Spectral Behaviour of a Simple Non-self-adjoint Operator

We investigate the spectrum of a typical non-selfadjoint differential operator AD = −d2/dx2 ⊗ A acting on L(0, 1) ⊗ C, where A is a 2 × 2 constant matrix. We impose Dirichlet and Neumann boundary conditions in the first and second coordinate respectively at both ends of [0, 1] ⊂ R. For A ∈ R we explore in detail the connection between the entries of A and the spectrum of AD, we find necessary c...

Weyl Function and Spectral Properties of Self-adjoint Extensions

Spectral properties of unbounded JJ-self-adjoint block operator matrices

We study the spectrum of unbounded J-self-adjoint block operator matrices. In particular, we prove enclosures for the spectrum, provide a sufficient condition for the spectrum being real and derive variational principles for certain real eigenvalues even in the presence of non-real spectrum. The latter lead to lower and upper bounds and asymptotic estimates for eigenvalues. AMS Subject classifi...

Spectral Theory for Compact Self-Adjoint Operators

This agrees with the definition of the spectrum in the matrix case, where the resolvent set comprises all complex numbers that are not eigenvalues. In terms of its spectrum, we will see that a compact operator behaves like a matrix, in the sense that its spectrum is the union of all of its eigenvalues and 0. We begin with the eigenspaces of a compact operator. We start with two lemmas that we w...