Inequalities for generalized derivations of operator monotone functions in norm ideals of compact operators

Abstract

Let Φ be a symmetrically norming (s.n.) function, p⩾2, Φ(p)⁎ to be a dual s.n. function to p-modified s.n. function Φ(p), ▪, with A and B being normal operators such that ▪. If both A and B are strictly accretive, then for non-constant Pick function φ∈P[0,+∞)(1)||φ(A)X−Xφ(B)||Φ⩽||φ′(A⁎+A2)(AX−XB)φ′(B+B⁎2)||Φ. If A and B have strictly contractive real parts, then12||I−|A⁎+A2|2(log⁡I+AI−AX−Xlog⁡I+BI−B)×I−|B+B⁎2|2||Φ⩽||AX−XB||Φ. If A is cohyponormal, B is hyponormal and at least one of them is normal, such that ▪, then(2)π2||cos⁡A⁎+Aπ(tan⁡2AπX−Xtan⁡2Bπ)cos⁡B+B⁎π||Φ(p)⁎⩽||AX−XB||Φ(p)⁎. Inequality (1) generalizes “difference” version of Heinz norm inequality [13, Hilfssatz 3] and mean values norm inequality [21, th. 4.4] for operator monotone functions. Inequality (2) remains valid for all s.n. function Φ if A and B are both (additionally) normal, which extends inequalities in [32, th. 5] and [34, rem. 25] for self-adjoint operators H and K, whence their spectra σ(H) and σ(K) are contained in (−π/2,π/2), to non necessarily self-adjoint operators A and B.