Isometric embeddability of S_q^m into S_pⁿ

In this paper, we study existence of isometric embedding of S_q^m into S_pⁿ, where 1≤p≠q≤∞ and n≥m≥2. We show that for all n≥m≥2 if there exists a linear isometry from S_q^m into S_pⁿ, where (q,p)∈(1,∞]×(1,∞)∪(1,∞)∖{3}×{1,∞} and p≠q, then we must have q=2. This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever S_q embeds isometrically into S_p for (q,p)∈(1,∞)×[2,∞)∪[4,∞)×{1}∪{∞}×(1,∞)∪[2,∞)×{∞} with p≠q, we must have q=2. Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative L_p-spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for m≥2 and 1<q<2, S_q^m embeds isometrically into S_∞ⁿ, was left open in Bull. London Math. Soc. 52 (2020) 437-447.