The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform
Vladimír Müller - One of the best experts on this subject based on the ideXlab platform.
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Quasisimilarity of power Bounded Operators and Blum–Hanson property☆
Journal of Functional Analysis, 2007Co-Authors: Vladimír Müller, Yuri TomilovAbstract:Abstract We construct a power Bounded Operator on a Hilbert space which is not quasisimilar to a contraction. To this aim, we solve an open problem from Operator ergodic theory showing that there are power Bounded Hilbert space Operators without the Blum–Hanson property. We also find an example of a power Bounded Operator quasisimilar to a unitary Operator which is not similar to a contraction, thus answering negatively open questions raised by Kerchy and Cassier. On the positive side, we prove that contractions on l p spaces ( 1 ⩽ p ∞ ) possess the Blum–Hanson property.
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Invariant subspaces for polynomially Bounded Operators
Journal of Functional Analysis, 2004Co-Authors: Călin Ambrozie, Vladimír MüllerAbstract:AbstractLet T be a polynomially Bounded Operator on a Banach space X whose spectrum contains the unit circle. Then T∗ has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions
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Power Bounded Operators and supercyclic vectors
Proceedings of the American Mathematical Society, 2003Co-Authors: Vladimír MüllerAbstract:We show that each power Bounded Operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the Operator has a nontrivial closed invariant homogeneous subset. Moreover, the Operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space Operators. For non-reflexive Banach spaces these results remain true: however, the non-supercyclic vector (invariant cone, respectively) relates to the adjoint of the Operator.
V. Müller - One of the best experts on this subject based on the ideXlab platform.
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Kreiss Bounded and uniformly Kreiss Bounded Operators
Revista Matemática Complutense, 2020Co-Authors: A. Bonilla, V. MüllerAbstract:If T is a Kreiss Bounded Operator on a Banach space, then $$\Vert T^n\Vert =O(n)$$ ‖ T n ‖ = O ( n ) . Forty years ago Shields conjectured that in Hilbert spaces, $$\Vert T^n\Vert = O(\sqrt{n})$$ ‖ T n ‖ = O ( n ) . A negative answer to this conjecture was given by Spijker, Tracogna and Welfert in 2003. We improve their result and show that this conjecture is not true even for uniformly Kreiss Bounded Operators. More precisely, for every $$\varepsilon >0$$ ε > 0 there exists a uniformly Kreiss Bounded Operator T on a Hilbert space such that $$\Vert T^n\Vert \sim (n+1)^{1-\varepsilon }$$ ‖ T n ‖ ∼ ( n + 1 ) 1 - ε for all $$n\in \mathbb {N}$$ n ∈ N . On the other hand, any Kreiss Bounded Operator on Hilbert spaces satisfies $$\Vert T^n\Vert =O(\frac{n}{\sqrt{\log n}})$$ ‖ T n ‖ = O ( n log n ) . We also prove that the residual spectrum of a Kreiss Bounded Operator on a reflexive Banach space is contained in the open unit disc, extending known results for power Bounded Operators. As a consequence we obtain examples of mean ergodic Hilbert space Operators which are not Kreiss Bounded.
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Kreiss Bounded and uniformly Kreiss Bounded Operators
arXiv: Functional Analysis, 2019Co-Authors: A. Bonilla, V. MüllerAbstract:If $T$ is a Kreiss Bounded Operator on a Banach space, then $\|T^n\|=O(n)$. Forty years ago Shields conjectured that in Hilbert spaces, $\|T^n\| = O(\sqrt{n})$. A negative answer to this conjecture was given by Spijker, Tracogna and Welfert in 2003. We improve their result and show that this conjecture is not true even for uniformly Kreiss Bounded Operators. More precisely, for every $\varepsilon>0$ there exists a uniformly Kreiss Bounded Operator $T$ on a Hilbert space such that $\|T^n\|\sim (n+1)^{1-\varepsilon}$ for all $n\in \Bbb N$. On the other hand, any Kreiss Bounded Operator on Hilbert spaces satisfies $\|T^n\|=O(\frac{n}{\sqrt{\log n}})$. We also prove that the residual spectrum of a Kreiss Bounded Operator on a reflexive Banach space is contained in the open unit disc, extending known results for power Bounded Operators. As a consequence we obtain examples of mean ergodic Hilbert space Operators which are not Kreiss Bounded.
Gilles Cassier - One of the best experts on this subject based on the ideXlab platform.
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Generalized Toeplitz Operators, restrictions to invariant subspaces and similarity problems.
Journal of Operator Theory, 2005Co-Authors: Gilles CassierAbstract:Our purpose is to investigate the asymptotic properties of an Operator T on an invariant subspace E 2 Lat(T) and on E? with the generalized Toeplitz Operators associated with T. We show how the relative properties may be used in order to give a general result linking the behaviour of T on E and on E? with the possibility for T to be similar to a scalar multiple of a contraction. Some applications are indicated. In particular, one of our results implies that there is no hope to construct a power Bounded Operator of Foguel type that is not similar to a contraction and such that for everyx in H-{0} the sequence (Tn) does not converge to 0. We also study the asymptotic and spectral properties of these Operators of Foguel type.
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On Power-Bounded Operators in Finite von Neumann Algebras
Journal of Functional Analysis, 1996Co-Authors: Gilles Cassier, Thierry FackAbstract:Abstract It is proved that any power-Bounded Operator of class C 1, · in a finite von Neumann algebra is conjugate to a unitary. This solves a conjecture stated by I. Kovacs in 1970. An important ingredient of the proof is the study of completely positive projections on some Operator space.
A. Bonilla - One of the best experts on this subject based on the ideXlab platform.
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Kreiss Bounded and uniformly Kreiss Bounded Operators
Revista Matemática Complutense, 2020Co-Authors: A. Bonilla, V. MüllerAbstract:If T is a Kreiss Bounded Operator on a Banach space, then $$\Vert T^n\Vert =O(n)$$ ‖ T n ‖ = O ( n ) . Forty years ago Shields conjectured that in Hilbert spaces, $$\Vert T^n\Vert = O(\sqrt{n})$$ ‖ T n ‖ = O ( n ) . A negative answer to this conjecture was given by Spijker, Tracogna and Welfert in 2003. We improve their result and show that this conjecture is not true even for uniformly Kreiss Bounded Operators. More precisely, for every $$\varepsilon >0$$ ε > 0 there exists a uniformly Kreiss Bounded Operator T on a Hilbert space such that $$\Vert T^n\Vert \sim (n+1)^{1-\varepsilon }$$ ‖ T n ‖ ∼ ( n + 1 ) 1 - ε for all $$n\in \mathbb {N}$$ n ∈ N . On the other hand, any Kreiss Bounded Operator on Hilbert spaces satisfies $$\Vert T^n\Vert =O(\frac{n}{\sqrt{\log n}})$$ ‖ T n ‖ = O ( n log n ) . We also prove that the residual spectrum of a Kreiss Bounded Operator on a reflexive Banach space is contained in the open unit disc, extending known results for power Bounded Operators. As a consequence we obtain examples of mean ergodic Hilbert space Operators which are not Kreiss Bounded.
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Kreiss Bounded and uniformly Kreiss Bounded Operators
arXiv: Functional Analysis, 2019Co-Authors: A. Bonilla, V. MüllerAbstract:If $T$ is a Kreiss Bounded Operator on a Banach space, then $\|T^n\|=O(n)$. Forty years ago Shields conjectured that in Hilbert spaces, $\|T^n\| = O(\sqrt{n})$. A negative answer to this conjecture was given by Spijker, Tracogna and Welfert in 2003. We improve their result and show that this conjecture is not true even for uniformly Kreiss Bounded Operators. More precisely, for every $\varepsilon>0$ there exists a uniformly Kreiss Bounded Operator $T$ on a Hilbert space such that $\|T^n\|\sim (n+1)^{1-\varepsilon}$ for all $n\in \Bbb N$. On the other hand, any Kreiss Bounded Operator on Hilbert spaces satisfies $\|T^n\|=O(\frac{n}{\sqrt{\log n}})$. We also prove that the residual spectrum of a Kreiss Bounded Operator on a reflexive Banach space is contained in the open unit disc, extending known results for power Bounded Operators. As a consequence we obtain examples of mean ergodic Hilbert space Operators which are not Kreiss Bounded.
Thierry Fack - One of the best experts on this subject based on the ideXlab platform.
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On Power-Bounded Operators in Finite von Neumann Algebras
Journal of Functional Analysis, 1996Co-Authors: Gilles Cassier, Thierry FackAbstract:Abstract It is proved that any power-Bounded Operator of class C 1, · in a finite von Neumann algebra is conjugate to a unitary. This solves a conjecture stated by I. Kovacs in 1970. An important ingredient of the proof is the study of completely positive projections on some Operator space.