Reduced commutativity of moduli of operators

Pietrzycki, Paweł

doi:10.1016/j.laa.2018.08.007

Cited by 7 publications

(8 citation statements)

References 32 publications

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“…To prove Theorem 1.3, we will need the following theorem, which is the second of the three main results of this paper. It generalizes [37,Lemma 3.7] in two directions. First, it removes the injectivity assumption, and second, it replaces commutativity by a more general intertwining relation.…”

Section: Theorem 11 [ [38 Theorem 12]] If T ∈ B(h ) Is a Subnormal Op...mentioning

confidence: 64%

On nth roots of bounded and unbounded quasinormal operators

Pietrzycki

Stochel

2022

Annali di Matematica

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In a recent paper (JFA 278:108342, 2020), R. E. Curto, S. H. Lee and J. Yoon asked the following question: LetTbe a subnormal operator, and assume that$$T^2$$ T 2 is quasinormal. Does it follow thatTis quasinormal? In (JFA 280:109001, 2021) we answered this question in the affirmative. In the present paper, we will extend this result in two directions. Namely, we prove that hyponormal (or even much beyond this class) nth roots of bounded quasinormal operators are quasinormal. On the other hand, we show that subnormal nth roots of unbounded quasinormal operators are quasinormal. We also prove that a non-normal quasinormal operator having a quasinormal nth root has a non-quasinormal nth root.

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Section: Theorem 11 [ [38 Theorem 12]] If T ∈ B(h ) Is a Subnormal Op...mentioning

confidence: 64%

On nth roots of bounded and unbounded quasinormal operators

Pietrzycki

Stochel

2022

Annali di Matematica

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“…Jab loński, I. B. Jung and J. Stochel realized the importance of this class as a vehicle to collect a number of interesting examples and counterexamples (see e.g., [5,9,[14][15][16][17]20,21,26]).…”

Section: Weighted Shifts On Rootless Directed Treesmentioning

confidence: 99%

Generalized Multipliers for Left-Invertible Operators and Applications

Pietrzycki

2020

Integr. Equ. Oper. Theory

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Generalized multipliers for a left-invertible operator T, whose formal Laurent series $$U_x(z)=\sum _{n=1}^\infty (P_ET^{n}x)\frac{1}{z^n}+\sum _{n=0}^\infty (P_E{T^{\prime *n}}x)z^n$$ U x ( z ) = ∑ n = 1 ∞ ( P E T n x ) 1 z n + ∑ n = 0 ∞ ( P E T ′ ∗ n x ) z n , $$x\in \mathcal {H}$$ x ∈ H actually represent analytic functions on an annulus or a disc are investigated. We show that they are coefficients of analytic functions and characterize the commutant of some left-invertible operators, which satisfies certain conditions in its terms. In addition, we prove that the set of multiplication operators associated with a weighted shift on a rootless directed tree lies in the closure of polynomials in z and $$\frac{1}{z}$$ 1 z of the weighted shift in the topologies of strong and weak operator convergence.

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“…implies the quasinormality of A. This problem has been studied by several authors (see [11,12,17,18,24,25]). Below we collect some results on this problem.…”

Section: Introductionmentioning

confidence: 99%

“…• A is bounded and satisfies (1.2) with S = {p, m, m + p, n, n + p} for some p, m, n ∈ N, m < n (see [18,Theorem 3.11]), then A is quasinormal.…”

Section: Introductionmentioning

confidence: 99%