Eigenvectors of Perturbed Operators

3 Spin Operators and Eigenvectors

Eigenvectors of Order-preserving Linear Operators

Suppose that K is a closed, total cone in a real Banach space X, that A :X!X is a bounded linear operator which maps K into itself, and that A« denotes the Banach space adjoint of A. Assume that r, the spectral radius of A, is positive, and that there exist x ! 1 0 and m& 1 with Am(x ! ) ̄ rmx ! (or, more generally, that there exist x ! a (®K ) and m& 1 with Am(x ! )& rmx ! ). If, in addition, A...

Lipschitz Functions of Perturbed Operators

We prove that if f is a Lipschitz function on R, A and B are self-adjoint operators such that rank(A − B) = 1, then f(A) − f(B) belongs to the weak space S1,∞, i.e., sj(A − B) ≤ const(1 + j). We deduce from this result that if A − B belongs to the trace class S1 and f is Lipschitz, then f(A) − f(B) ∈ SΩ, i.e., Pn j=0 sj(f(A) − f(B)) ≤ const log(2 + n). We also obtain more general results about ...

On Eigenvectors of Nilpotent Lie Algebras of Linear Operators

We give a condition ensuring that the operators in a nilpotent Lie algebra of linear operators on a finite dimensional vector space have a common eigenvector. Introduction Throughout this paper V is a vector space of positive dimension over a field f and g is a nilpotent Lie algebra over f of linear operators on V . An element u ∈ V is an eigenvector for S ⊂ g if u is an eigenvector for every o...

Functions of Perturbed Noncommuting Self-adjoint Operators

Abstract. We consider functions f(A,B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B ∞,1 (R), then we have the following Lipschitz type estimate in the trace norm: ‖f(A1, B1)− f(A2, B2)‖S1 ≤ const(‖A1 −A2‖S1 + ‖B1 −B2‖S1). However, the condition f ∈ B ∞,1 (R) does not imply the Lipschitz ...