On Isometric Embeddability of $S_q^m$ into $S_p^n$ as non-commutative Quasi-Banach spaces

Abstract

The existence of isometric embedding of S m q into S n p , where 1 ≤ p = q ≤ ∞ and m, n ≥ 2 has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above mentioned range of p and q. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space ℓ m q (R) into ℓ n p (R), where (q, p) ∈ (0, ∞) × (0, 1) and p = q. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of S m q into S n p , where (q, p) ∈ (0, 2) × (0, 1) ∪ {1} × (0, 1) \ { 1 n : n ∈ N} ∪ {∞} × (0, 1) \ { 1 n : n ∈ N} and p = q. Moreover, in some restrictive cases, we also show that there is no isometric embedding of S m q into S n p , where (q, p) ∈ [2, ∞) × (0, 1). To achieve our goal we significantly use Kato-Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.