Preservation of the Unconditional Basis Property under Non-Self-Adjoint Perturbations of Self-Adjoint Operators

Non-negative Perturbations of Non-negative Self-adjoint Operators

Let A be a non-negative self-adjoint operator in a Hilbert space H and A0 be some densely defined closed restriction of A0, A0 ⊆ A 6= A0. It is of interest to know whether A is the unique non-negative self-adjoint extensions of A0 in H. We give a natural criterion that this is the case and if it fails, we describe all non-negative extensions of A0. The obtained results are applied to investigat...

Rank one perturbations of self-adjoint operators

Non-self-adjoint Differential Operators

We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...

On the computation of spectra and pseudospectra of self-adjoint and non-self-adjoint Schrödinger operators

We consider the long standing open question on whether one can actually compute spectra and pseudospectra of arbitrary (possibly non-self-adjoint) Schrödinger operators.We conclude that the answer is affirmative for “almost all” such operators, meaning that the operators must satisfy rather weak conditions such as the spectrum cannot be empty nor the whole plane. We include algorithms for the g...

Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let A : D(A) ⊆ H → H be an injective self-adjoint operator and let τ : D(A) → X, X a Banach space, be a surjective linear map such that ‖τφ‖X ≤ c ‖Aφ‖H. Supposing that Range (τ ) ∩ H = {0}, we define a family AτΘ of self-adjoint operators which are extensions of the symmetric operator A|{τ=0} . Any φ in the operator domain D(A τ Θ) is characterized by a sort of boundary conditions on its univoc...