New Counterexamples on Ritt Operators, Sectorial Operators and -Boundedness

Arnold, Loris; Merdy, Christian Le

doi:10.1017/s0004972719000431

Cited by 4 publications

(2 citation statements)

References 16 publications

(23 reference statements)

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“…The proof in the case 2 < p < ∞ is similar. In [2] we construct a Ritt operator such that the set {T n : n ∈ N} is not γ-bounded (in particular T is not γ-Ritt).…”

Section: Power γ-Bounded Operatorsmentioning

confidence: 99%

Derivative bounded functional calculus of power bounded operators on Banach spaces

Arnold¹

2020

Preprint

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In this article we study bounded operators T on Banach space X which satisfy the discrete Gomilko Shi-Feng conditionWe show that it is equivalent to a certain derivative bounded functional calculus and also to a bounded functional calculus relative to Besov space. Also on Hilbert space discrete Gomilko Shi-Feng condition is equivalent to power-boundedness. Finally we discuss the last equivalence on general Banach space involving the concept of γ-boundedness.

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“…The proof in the case 2 < p < ∞ is similar. In [2] we construct a Ritt operator such that the set {T n : n ∈ N} is not γ-bounded (in particular T is not γ-Ritt).…”

Section: Power γ-Bounded Operatorsmentioning

confidence: 99%

Derivative bounded functional calculus of power bounded operators on Banach spaces

Arnold¹

2020

Preprint

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“…Our proof will basically be a reconstruction of the idea of Lancien and the first author [KL00] to construct sectorial operators that are not Rsectorial. This idea has been further developed in a sequence of papers by Fackler [Fac13,Fac14a,Fac15,Fac16] and was recently revisited by Arnold and Le Merdy [AL19].…”

Section: Sectorial Operators Which Are Not Almost α-Sectorialmentioning

confidence: 99%

Euclidean structures and operator theory in Banach spaces

Kalton,

Lorist,

Weis

2019

Preprint

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We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ on a Hilbert space. Our assumption on Γ is expressed in terms of α-boundedness for a Euclidean structure α on the underlying Banach space X. This notion is originally motivated by R-or γ-boundedness of sets of operators, but for example any operator ideal from the Euclidean space ℓ 2 n to X (like the γradonifying or the 2-summing operator ideal) defines such a structure. Therefore our method is quite general and flexible and allows to unify the approach to seemingly unrelated theorems. Conversely we show that Γ has to be α-bounded for some Euclidean structure α for it to be representable on a Hilbert space.By choosing the Euclidean structure α accordingly we get a unified and more general approach to classical factorization theorems like the Kwapień-Maurey factorization theorem, an improved version of the Banach function space-valued extension theorem of Rubio de Francia, a quantitative proof of the boundedness of the lattice Hardy-Littlewood maximal operator and the equivalence of the UMD and the dyadic UMD + property on Banach function spaces. Furthermore we use these Euclidean structures to build vector-valued function spaces, which enjoy the nice property that any bounded operator on L 2 extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method.Using our representation theorem we prove a quite general transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We extend and refine the known theory based on R-boundedness for the joint and operator-valued H ∞ -calculus and the "sum of operators" theorem for commuting sectorial operators. Moreover we extend the classical characterization of the boundedness of the H ∞ -calculus on Hilbert spaces in terms of BIP, square functions and dilations to the Banach space setting. Furthermore we establish via the H ∞ -calculus a version of Littlewood-Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the Banach space geometry of X, such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 on a closed subspace of L p for 1 < p < ∞ with a bounded H ∞ -calculus with optimal angle ωH∞ (A) > 0.

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Euclidean Structures and Operator Theory in Banach Spaces

Kalton¹,

Lorist²,

Weis³

2023

Memoirs of the AMS

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We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ \Gamma on a Hilbert space. Our assumption on Γ \Gamma is expressed in terms of α \alpha -boundedness for a Euclidean structure α \alpha on the underlying Banach space X X . This notion is originally motivated by R \mathcal {R} - or γ \gamma -boundedness of sets of operators, but for example any operator ideal from the Euclidean space ℓ n 2 \ell ^2_n to X X defines such a structure. Therefore, our method is quite flexible. Conversely we show that Γ \Gamma has to be α \alpha -bounded for some Euclidean structure α \alpha to be representable on a Hilbert space. By choosing the Euclidean structure α \alpha accordingly, we get a unified and more general approach to the Kwapień–Maurey factorization theorem and the factorization theory of Maurey, Nikišin and Rubio de Francia. This leads to an improved version of the Banach function space-valued extension theorem of Rubio de Francia and a quantitative proof of the boundedness of the lattice Hardy–Littlewood maximal operator. Furthermore, we use these Euclidean structures to build vector-valued function spaces. These enjoy the nice property that any bounded operator on L 2 L^2 extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem, we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We for example extend and refine the known theory based on R \mathcal {R} -boundedness for the joint and operator-valued H ∞ H^\infty -calculus. Moreover, we extend the classical characterization of the boundedness of the H ∞ H^\infty -calculus on Hilbert spaces in terms of B I P BIP , square functions and dilations to the Banach space setting. Furthermore we establish, via the H ∞ H^\infty -calculus, a version of Littlewood–Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the geometry of X X , such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 0 on a closed subspace of L p L^p for 1 > p > ∞ 1>p>\infty with a bounded H ∞ H^\infty -calculus with optimal angle ω H ∞ ( A ) > 0 \omega _{H^\infty }(A) >0 .

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New Counterexamples on Ritt Operators, Sectorial Operators and -Boundedness

Cited by 4 publications

References 16 publications

Derivative bounded functional calculus of power bounded operators on Banach spaces

Derivative bounded functional calculus of power bounded operators on Banach spaces

Euclidean structures and operator theory in Banach spaces

Euclidean Structures and Operator Theory in Banach Spaces

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