2010/08/24 by Filonov, Nikolay, Kachkovskiy, Ilya
#FOS: Mathematics #Operator Algebras (math.OA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.1008.4002
The paper is devoted to the following question: consider two self-adjoint n× n-matrices H1,H2, ‖H1‖≤ 1, ‖H2‖≤ 1, such that their commutator [H1,H2] is small in some sence. Do there exist such self-adjoint commuting matrices A1,A2, such that Ai is close to Hi, i=1,2? The answer to this question is positive if the smallness is considered with respect to the operator norm. The following result was established by Huaxin Lin: if ‖[H1,H2]‖=δ, then we can choose Ai such that ‖Hi-Ai‖≤ C(δ), i=1,2, where C(δ)→ 0 as δ→ 0. Notice that C(δ) does not depend on n. The proof was simplified by Friis and Rørdam. A quantitative version of the result with C(δ)=E(1/δ)δ1/5, where E(x) grows slower than any power of x, was recently established by Hastings. We are interested in the same question, but with respect to the normalized Hilbert-Schmidt norm. An analog of Lin's theorem for this norm was established by Hadwin and independently by Filonov and Safarov. A quantitative version with C(δ)=12δ1/6, where δ=‖[H1,H2]‖\tr, was recently obtained by Glebsky. In the present paper, we use the same ideas to prove a similar result with C(δ)=2δ1/4. We also refine Glebsky's theorem concerning the case of n operators.