scholarly journals Boundedly Spaced Subsequences and Weak Dynamics

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
C. S. Kubrusly ◽  
P. C. M. Vieira
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Bounded Operator ◽  
Weak Stability ◽  
Weakly Stable ◽  
Power Bounded Operator ◽  
Power Bounded

Weak supercyclicity is related to weak stability, which leads to the question that asks whether every weakly supercyclic power bounded operator is weakly stable. This is approached here by investigating weak l-sequential supercyclicity for Hilbert-space contractions through Nagy–Foliaş–Langer decomposition, thus reducing the problem to the quest of conditions for a weakly l-sequentially supercyclic unitary operator to be weakly stable, and this is done in light of boundedly spaced subsequences.

scholarly journals On power-bounded prespectral operators

1976 ◽  
Vol 20 (2) ◽  
pp. 173-175
Author(s):  
H. R. Dowson
Keyword(s):  
Growth Condition ◽  
Bounded Operator ◽  
Spectral Operator ◽  
Scalar Type ◽  
Image Position ◽  
Power Bounded Operator ◽  
Power Bounded

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form

scholarly journals Some notes on complex symmetric operators

2021 ◽  
Vol 2021 (1) ◽  
pp. 90-96
Author(s):  
Marcos S. Ferreira
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Polar Decomposition ◽  
Bounded Operator ◽  
Complex Symmetric ◽  
Symmetric Operators

Abstract In this paper we show that every conjugation C on the Hardy-Hilbert space H 2 is of type C = T * 𝒥T, where T is an unitary operator and 𝒥 f ( z ) = f ( z ¯ ) ¯ \mathcal{J}f\left( z \right) = \overline {f\left( {\bar z} \right)} with f ∈ H 2. Moreover we prove some relations of complex symmetry between the operators T and |T|, where T = U |T| is the polar decomposition of bounded operator T ∈ ℒ(ℋ) on the separable Hilbert space ℋ.

scholarly journals Mean ergodic operators and reflexive Fréchet lattices

2011 ◽  
Vol 141 (5) ◽  
pp. 897-920 ◽  
Author(s):  
José Bonet ◽  
Ben de Pagter ◽  
Werner J. Ricker
Keyword(s):  
Bounded Operator ◽  
Banach Lattices ◽  
Positive Power ◽  
Bounded Linear ◽  
Mean Ergodic Operators ◽  
Power Bounded Operator ◽  
Complemented Lattice ◽  
Locally Convex ◽  
Strong Dual ◽  
Power Bounded

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E is Dedekind σ-complete and every positive power-bounded operator on E is mean ergodic if and only if every positive power-bounded operator in the strong dual E′β (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c0, l1 or l∞.

Factorization of an adjontable Markov operator

2019 ◽  
Vol 22 (02) ◽  
pp. 1950013
Author(s):  
Carlo Pandiscia
Keyword(s):  
Hilbert Space ◽  
Matrix Algebra ◽  
Unitary Operator ◽  
Markov Operator ◽  
Factorization Property ◽  
Markov Operators ◽  
Dilation Theory ◽  
Probability Spaces

In this work, we propose a method to investigate the factorization property of a adjontable Markov operator between two algebraic probability spaces without using the dilation theory. Assuming the existence of an anti-unitary operator on Hilbert space related to Stinespring representations of our Markov operator, which satisfy some particular modular relations, we prove that it admits a factorization. The method is tested on the two typologies of maps which we know admits a factorization, the Markov operators between commutative probability spaces and adjontable homomorphism. Subsequently, we apply these methods to particular adjontable Markov operator between matrix algebra which fixes the diagonal.

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

On Products of Symmetries

1966 ◽  
Vol 18 ◽  
pp. 897-900 ◽  
Author(s):  
Peter A. Fillmore
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Type Ii ◽  
Central Projections ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Principal Tool ◽  
The Subject

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebraA(i.e., one with no finite non-zero central projections), then the symmetries may be chosen fromA.A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.

scholarly journals Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

1968 ◽  
Vol 32 ◽  
pp. 141-153 ◽  
Author(s):  
Masasi Kowada
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Spectral Type ◽  
Measure Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Unitary Operators ◽  
Parameter Group ◽  
Probability Measure Space

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators {Ut} on a separable Hilbert space H, and discuss their spectral types, although U and {Ut} are not necessarily supposed to be derived from an automorphism or a flow respectively.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

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