Some results on higher order isometries and symmetries: Products and sums with a nilpotent operator

Gu, Caixing; Stankus, Mark

doi:10.1016/j.laa.2014.09.044

Cited by 29 publications

(17 citation statements)

References 13 publications

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“…In [2] the connection between Jordan operators and 3-symmetric operators was established in general. A sample of the more recent articles related to 3-isometries includes [3][4][5][6][7][8][9][10][11]15,16,18,20].…”

Section: An Operator T ∈ B(h) Is a 3-isometry Ifmentioning

confidence: 99%

The 3-Isometric Lifting Theorem

McCullough

Russo

2015

Integr. Equ. Oper. Theory

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An operator T on Hilbert space is a 3-isometry if T * n T n = I + nB1 + n 2 B2 is quadratic in n. An operator J is a Jordan operator if J = U + N where U is unitary, N 2 = 0 and U and N commute. If T is a 3-isometry and c > 0, then I − c −2 B2 + sB1 + s 2 B2 is positive semidefinite for all real s if and only if it is the restriction of a Jordan operator J = U + N with the norm of N at most c. As a corollary, an analogous result for 3-symmetric operators, due to Helton and Agler, is recovered.Mathematics Subject Classification. 47A20 (Primary), 47A45, 47B99, 34B24 (Secondary).

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Section: An Operator T ∈ B(h) Is a 3-isometry Ifmentioning

confidence: 99%

The 3-Isometric Lifting Theorem

McCullough

Russo

2015

Integr. Equ. Oper. Theory

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“…Motivated by [9] and [19], in Section 2 we show that if S is a left m-inverse of T and Q is a nilpotent operator of order l commuting with S, then S + Q is a left n-inverse of T where n = m + l − 1. We also discuss the converse of this result.…”

mentioning

confidence: 99%

“…Finally, we acknowledge that some ideas and techniques are borrowed without explicit mention from the author's paper [18] and from the author and Stankus's paper [19] where related questions and more for n-isometries and n-symmetric operators are studied. But this paper is self-contained and will focus on left n-invertible operators.…”

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confidence: 99%

Structures of left n-invertible operators and their applications

2015

Studia Math.

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We study left n-invertible operators introduced in two recent papers. We show how to construct a left n-inverse as a sum of a left inverse and a nilpotent operator. We provide refinements for results on products and tensor products of left n-invertible operators by Duggal and Müller (2013). Our study leads to improvements and different and often more direct proofs of results of Duggal and Müller (2013) and Sid Ahmed (2012). We make a conjecture about tensor products of left n-invertible operators and prove this conjecture in several cases. Finally, applications of these results are given to left n-invertible elementary operators and essentially left n-invertible operators.

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“…Moreover, it is shown in [2] that if A is an m-isometry then A + Q is a (2N − m − 2)-isometry. Recently, such operators have been considered by several authors; for example, see [4,6] .…”

Section: Let H Be a Hilbert Space And B(h) Stands For The Space Of Almentioning

confidence: 99%