2023/11/29 by Herrera, David
#15A60 #26A15 #26A16 #26A51 #47A30 (Primary) 15A45 #47A63 #47B15 #47B47 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2311.17448
Let f ≥ 0 be operator monotone on [0, ∞). In this paper we prove that for any unitarily-invariant norm |||-||| on Mn(ℂ) and matrices A, B, X ∈ Mn(ℂ) with A, B ≥ 0 and |||X||| ≤ 1, |||f(A)X-Xf(B)||| ≤ C f(|||AX-XB|||) for C < 1.01975. We do this by reducing this inequality to a function approximation problem and we choose approximate minimizers. This is much progress toward the conjecturally optimal value of C=1 which is known only in the case of the Hilbert-Schmidt norm. When |||-||| is the the operator norm ||-||, we obtain a great reduction of the previously known estimate of C = 1.25. We further prove that for |||X||| ≤ 1, |||A1/2X-XB1/2||| ≤ 1.00891 |||AX-XB|||. This is a great improvement toward the conjecture of G. Pedersen that this inequality for |||-||| being the operator norm holds with C = 1. We discuss other related inequalities, including some sharp commutator inequalities. We also prove a sharp equivalence inequality between the operator modulus of continuity and the commutator modulus of continuity for continuous functions on ℝ.