On a subspace perturbation problem

• Published in: Proceedings of the American Mathematical Society Vol. 131; no. 11; pp. 3469 - 3476
• Main Authors: Kostrykin, Vadim, Makarov, Konstantin A., Motovilov, Alexander K.
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 01-11-2003
• Subjects: Borel sets; College mathematics; Eigenvalues; Hilbert spaces; Mathematical theorems; Mathematics; Perturbation theory; Separable spaces; Sine function; Spectral theory; Perturbation theory; spectral subspaces
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-03-06917-X

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let A and V be bounded self-adjoint operators. Assume that the spectrum of A consists of two disjoint parts \sigma and \Sigma such that d=\text{dist}(\sigma, \Sigma)>0. We show that the norm of the difference of the spectral projections \[\mathsf{E}_A(\sigma)\quad \text{and} \quad \mathsf{E}_{A+V}\big (\{\lambda | \dist(\lambda, \sigma)<d/2\}\big)\] for A and A+V is less than one whenever either (i) \|V\|<\frac{2}{2+\pi}d or (ii) \|V\|<\frac{1}{2}d and certain assumptions on the mutual disposition of the sets \sigma and \Sigma are satisfied.