Symmetric norms and spaces of operators
Symmetric norms and spaces of operators
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DOI：
10.1007/978-3-319-18796-9_15
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发表时间：
2015
影响因子：
3.5
通讯作者：
F. Gesztesy;G. Godefroy;Loukas Grafakos;I. Verbitsky
中科院分区：
文献类型：
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作者：
F. Gesztesy;G. Godefroy;Loukas Grafakos;I. Verbitsky
我们证明，如果 (E, II· II£) 是对称 Banach 序列空间，则由 T E Y£ 定义的可分离希尔伯特空间上的算子对应空间 YE 当且仅当 (s,(T)),': 1 E£ 是范数 II Tll9'£= II (s,(T)):, 11£ 下的 Banach 空间。尽管冯·诺依曼于 1937 年在有限维空间中证明了这一点，但它从未在无限维空间中完全普遍化；先前的证明使用了 E 上完全对称性的更强假设。II·119'£ 是范数的证明需要均匀 Hardy-Littlewood 大化的明显新概念；完整性还需要新的证明。我们还给出了基于具有正常忠实半有限迹的半有限冯诺依曼代数建模的算子空间的类似结果。
We show that if (E, II· II£) is a symmetric Banach sequence space then the corresponding space YE of operators on a separable Hilbert space, defined by T E Y£ if and only if (s,(T)),': 1 E£, is a Banach space under the norm II Tll9'£= II (s,(T)):, 11£. Although this was proved for finite-dimensional spaces by von Neumann in 1937, it has never been established in complete generality in infinite-dimensional spaces; previous proofs have used the stronger hypothesis of full symmetry on E. The proof that II· 119'£ is a norm requires the apparently new concept of uniform Hardy-Littlewood majorization; completeness also requires a new proof. We also give the analogous results for operator spaces modelled on a semifinite von Neumann algebra with a normal faithful semi-finite trace.