A generalized Fuglede-Putnam theorem for a class of almost normal tuples of operators with finite modulus of Hilbert-Schmidt quasitriangularity. (English) Zbl 1474.47058
Let \(H\) be a separable, infinite-dimensional complex Hilbert space, and let \(S = (S^1,S^2,\ldots,S^n)\) be an almost normal tuple of linear operators acting on \(H\). Assume that there exists an operator \(X\) such that \(\Delta =\sum_{j=1}^n S^j X S^j\) is Hilbert-Schmidt. Then \(\Delta^\ast = \sum_{j=1}^n S^{\ast j} X S^{\ast j}\) is also Hilbert-Schmidt, and there exists a precise affine bound between the two, involving the moduli of quasitriangularities of \(S\) and \(S^\ast\).
This is a major generalization of prior results, in particular, G. Weiss [Trans. Am. Math. Soc. 278, 1–20 (1983; Zbl 0532.47013)] and V. Shulman and L. Turowska [J. Reine Angew. Math. 590, 143–187 (2006; Zbl 1094.47054)].
This is a major generalization of prior results, in particular, G. Weiss [Trans. Am. Math. Soc. 278, 1–20 (1983; Zbl 0532.47013)] and V. Shulman and L. Turowska [J. Reine Angew. Math. 590, 143–187 (2006; Zbl 1094.47054)].
Reviewer: Mihai Putinar (Santa Barbara)
MSC:
| 47B20 | Subnormal operators, hyponormal operators, etc. |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
| 47B47 | Commutators, derivations, elementary operators, etc. |
