A Course in Operator Theory

Conway, John B.

doi:10.1090/gsm/021

Cited by 295 publications

(330 citation statements)

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“…See for example [16]. Using this, Theorem 7.8 and the fact that recurrence is preserved by unitary equivalence we immediately get the following corollary.…”

Section: Proposition 81 If a Normal Operator T ∶ H → H Is Recurrentmentioning

confidence: 65%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

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Abstract. We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions. The most studied notion in linear dynamics is that of hypercyclicity: a bounded linear operator T acting on a separable Banach space is hypercyclic if there exists a vector whose orbit under T is dense in the space. On the other hand, a very central notion in topological dynamics is that of recurrence. This notion goes back to Poincaré and Birkhoff and it refers to the existence of points in the space for which parts of their orbits under a continuous map "return" to themselves. The purpose of this note is the study of the notion of recurrence, together with its variations, in the context of linear dynamics. Some examples and characterizations of recurrence for special In an effort to characterize recurrent linear operators one many times falls back to the notion of hypercyclicity. This is for example the case when we study the recurrence properties of backwards shifts, say on ℓ 2 (Z). The reason behind is that, according to a result of Seceleanu, [51], the orbits of these operators satisfy a zero-one law: if the orbit of a weighted backward shift contains a non-zero limit point then the corresponding shift is actually hypercyclic. Thus a weighted backward shift on ℓ 2 (Z) is recurrent if and only if it is hypercyclic. The same equivalence is true, albeit for different reasons, for the adjoint of a multiplication operators on the Hardy space H 2 (D). These connections to hypercyclicity, already observed in [17], come up naturally and thus motivate a further search on whether the properties of recurrent operators resemble the properties of hypercyclic ones, in general. It turns out that, indeed, there are many structural similarities between the set of hypercyclic vectors and the set of recurrent vectors in the sense that they exhibit the same invariances. Furthermore, the spectral properties of hypercyclic and recurrent operators are somewhat similar, although this vague statement should be interpreted with some care. However, these similarities cannot be pushed too much as there are obviously many classes of operators which are recurrent without being hypercyclic. One can find such examples among composition operators on the Hardy space H 2 (D). However, the primordial example is given just by considering unimodular multiples of the identity operator. A more general class for which one needs to address the recurrence properties independently of hypercyclicity is that of unitary operators on Hilbert spaces.The discussion above hopefully justifies why we will shortly recall a full set of definitions rel...

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“…See for example [16]. Using this, Theorem 7.8 and the fact that recurrence is preserved by unitary equivalence we immediately get the following corollary.…”

Section: Proposition 81 If a Normal Operator T ∶ H → H Is Recurrentmentioning

confidence: 65%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

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“…Namely, let a be normal and f a Schur function on σ(a). Given ε > 0, by the Weyl-von Neumann-Bergh theorem [11,Corollary 39.6] there exists a diagonal operator a 0 such that σ(a 0 ) ⊆ σ(a), a − a 0 < ε and (approximating f by polynomials)…”

Section: Commutators Of Functions Of Subnormal Operatorsmentioning

confidence: 99%

Variance of operators and derivations

Magajna

2016

Journal of Mathematical Analysis and Applications

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Abstract. The variance of a bounded linear operator a on a Hilbert space H at a unit vector ξ is defined by D ξ (a) = aξ 2 − | aξ, ξ | 2 . We show that two operators a and b have the same variance at all vectors ξ ∈ H if and only if there exist scalars σ, λ ∈ C with |σ| = 1 such that b = σa + λ1 or a is normal and b = σa * + λ1. Further, if a is normal, then the inequality D ξ (b) ≤ κD ξ (a) holds for some constant κ and all unit vectors ξ if and only if b = f (a) for a Lipschitz function f on the spectrum of a. Variants of these results for C * -algebras are also proved, where vectors are replaced by pure states.We also study the related inequalities bx − xb ≤ ax − xa supposed to hold for all x ∈ B(H) or for all x ∈ B(H n ) and all n ∈ N. We consider the connection between such inequalities and the range inclusion If a is subnormal, we study these conditions in particular in the case when b is of the form b = f (a) for a function f .

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“…In this section we shall need an extension of Arveson's version of Voiculescu's theorem [1,Theorem 4] to the case of bimodule maps over certain von Neumann algebras (Theorem 3.2 below). The proof of this theorem relies on the techniques developed in [1] and we shall give below only the necessary adjustments, referring to [1] or [5] for the details. (We have not been able to deduce the result from already known variants of Voiculescu's theorem [17] or [18] since no condition of nuclearity or exactness is present in our situation.)…”

Section: The Density In the Point Norm Topology Of Ncb A (B(h))mentioning

confidence: 99%

On completely bounded bimodule maps over W^*-algebras

Magajna

2003

Studia Math.

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Abstract. It is proved that for a von Neumann algebra A ⊆ B(H) the subspace of normal maps is dense in the space of all completely bounded A-bimodule homomorphisms of B(H) in the point norm topology if and only if the same holds for the corresponding unit balls, which is the case if and only if A is atomic with no central summands of type I ∞,∞ . Then a duality result for normal operator modules is presented and applied to the following problem. Given an operator space X and a von Neumann algebra A, is the map, from the extended Haagerup tensor product to the normal version of the Pisier delta tensor product a quotient map? We give a reformulation of this problem in terms of normal extension of some completely bounded maps and answer it affirmatively in the case A is of type I and X belongs to a certain class which includes all finite-dimensional operator spaces.

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A Course in Operator Theory

Cited by 295 publications

References 0 publications

Recurrent Linear Operators

Recurrent Linear Operators

Variance of operators and derivations

On completely bounded bimodule maps over W^*-algebras

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A Course in Operator Theory

Cited by 295 publications

References 0 publications

Recurrent Linear Operators

Recurrent Linear Operators

Variance of operators and derivations

On completely bounded bimodule maps over W*-algebras

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On completely bounded bimodule maps over W^*-algebras