Abstract: We establish the equivalence between two standard ways of defining symmetric sequence spaces. One approach, common in the theory of rearrangement-invariant spaces, requires invariance under all coordinate permutations together with solidity with respect to the lattice order. The other, used in the classical framework of Krein, Petunin and Semenov, is formulated in terms of the decreasing rearrangement of a sequence. We prove that, for every \(p\)-normed sequence space contained in \(c_0\), these two definitions determine exactly the same class of spaces. The argument applies throughout the range \(0 < p\leq1\). We also explain why the assumption \(E\subset c_0\) is natural: outside \(c_0\), the only rearrangement-invariant sequence space is \(\ell_\infty\), up to equivalence of norms. Finally, we discuss the corresponding terminology for ideals of compact operators and its connection with unitarily invariant norms, thereby clarifying the passage from sequence spaces to operator ideals in this discrete setting.
Funding: Y. Nessipbayev was partially supported by the grant no.~AP26102625 of the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan. F. Sukochev was partially supported by the Australian Research Council (DP230100434).
For citation: Zanin, D. V., Nessipbayev, Y. Kh. and Sukochev, F. A. Symmetric and Rearrangement-Invariant Sequence Spaces: a Unified Framework, Vladikavkaz Math. J., 2026, vol. 28, no. 2, pp.70-79. DOI 10.46698/j5915-0785-9974-a
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