Reducing subspaces for the product of a forward and a backward operator-weighted shifts

Xu Tang; Caixing Gu; Yufeng Lu; Yanyue Shi

doi:10.1016/j.jmaa.2021.125206

Reducing subspaces for the product of a forward and a backward operator-weighted shifts

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2021.125206 ◽

2021 ◽

Vol 501 (2) ◽

pp. 125206

Author(s):

Xu Tang ◽

Caixing Gu ◽

Yufeng Lu ◽

Yanyue Shi

Keyword(s):

Weighted Shifts ◽

Reducing Subspaces

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REDUCING SUBSPACES OF WEIGHTED SHIFTS WITH OPERATOR WEIGHTS

Bulletin of the Korean Mathematical Society ◽

10.4134/bkms.b150774 ◽

2016 ◽

Vol 53 (5) ◽

pp. 1471-1481 ◽

Cited By ~ 2

Author(s):

Caixing Gu

Keyword(s):

Weighted Shifts ◽

Reducing Subspaces

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Reducing subspaces of tensor products of weighted shifts

Science China Mathematics ◽

10.1007/s11425-015-5089-y ◽

2015 ◽

Vol 59 (4) ◽

pp. 715-730 ◽

Cited By ~ 3

Author(s):

KunYu Guo ◽

XuDi Wang

Keyword(s):

Tensor Products ◽

Weighted Shifts ◽

Reducing Subspaces

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Unitary equivalence and reductibility or invertibly weighted shifts

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270004702x ◽

1971 ◽

Vol 5 (2) ◽

pp. 157-173 ◽

Cited By ~ 13

Author(s):

Alan Lambert

Keyword(s):

Hilbert Space ◽

Sufficient Conditions ◽

Bounded Sequence ◽

Complex Hilbert Space ◽

Weighted Shifts ◽

Infinite Dimensional ◽

Reducing Subspaces ◽

Positive Weights ◽

Completely Reducible ◽

Necessary And Sufficient

Let H be a complex Hilbert space and let {A1, A2, …} be a uniformly bounded sequence of invertible operators on H. The operator S on l2(H) = H ⊕ H ⊕ … given by S〈x0, x1, …〉 = 〈0, A1x0, A2x1, …〉 is called the invertibly veighted shift on l2(H) with weight sequence {An }. A matricial description of the commutant of S is established and it is shown that S is unitarily equivalent to an invertibly weighted shift with positive weights. After establishing criteria for the reducibility of S the following result is proved: Let {B1, B2, …} be any sequence of operators on an infinite dimensional Hilbert space K. Then there is an operator T on K such that the lattice of reducing subspaces of T is isomorphic to the corresponding lattice of the W* algebra generated by {B1, B2, …}. Necessary and sufficient conditions are given for S to be completely reducible to scalar weighted shifts.

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