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More classes of non-orbit-transitive operators

Carl Pearcy, Lidia Smith (2010)

Studia Mathematica

In [JKP] and its sequel [FPS] the authors initiated a program whose (announced) goal is to eventually show that no operator in ℒ(ℋ) is orbit-transitive. In [JKP] it is shown, for example, that if T ∈ ℒ(ℋ) and the essential (Calkin) norm of T is equal to its essential spectral radius, then no compact perturbation of T is orbit-transitive, and in [FPS] this result is extended to say that no element of this same class of operators is weakly orbit-transitive. In the present note we show that no compact...

Non-hyperreflexive reflexive spaces of operators

Roman V. Bessonov, Janko Bračič, Michal Zajac (2011)

Studia Mathematica

We study operators whose commutant is reflexive but not hyperreflexive. We construct a C₀ contraction and a Jordan block operator S B associated with a Blaschke product B which have the above mentioned property. A sufficient condition for hyperreflexivity of S B is given. Some other results related to hyperreflexivity of spaces of operators that could be interesting in themselves are proved.

On a decomposition for pairs of commuting contractions

Zbigniew Burdak (2007)

Studia Mathematica

A new decomposition of a pair of commuting, but not necessarily doubly commuting contractions is proposed. In the case of power partial isometries a more detailed decomposition is given.

On class A operators

Sungeun Jung, Eungil Ko, Mee-Jung Lee (2010)

Studia Mathematica

We show that every class A operator has a scalar extension. In particular, such operators with rich spectra have nontrivial invariant subspaces. Also we give some spectral properties of the scalar extension of a class A operator. Finally, we show that every class A operator is nonhypertransitive.

On Deddens΄s Theorem

S. Giotopoulos (1981)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

On invariant subspaces for polynomially bounded operators

Junfeng Liu (2017)

Czechoslovak Mathematical Journal

We discuss the invariant subspace problem of polynomially bounded operators on a Banach space and obtain an invariant subspace theorem for polynomially bounded operators. At the same time, we state two open problems, which are relative propositions of this invariant subspace theorem. By means of the two relative propositions (if they are true), together with the result of this paper and the result of C. Ambrozie and V. Müller (2004) one can obtain an important conclusion that every polynomially...

On operator bands

Roman Drnovšek, Leo Livshits, Gordon MacDonald, Ben Mathes, Heydar Radjavi, Peter Šemrl (2000)

Studia Mathematica

A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space l 2 which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on l 2 that is weakly...

On reflexive subobject lattices and reflexive endomorphism algebras

Dong Sheng Zhao (2003)

Commentationes Mathematicae Universitatis Carolinae

In this paper we study the reflexive subobject lattices and reflexive endomorphism algebras in a concrete category. For the category Set of sets and mappings, a complete characterization for both reflexive subobject lattices and reflexive endomorphism algebras is obtained. Some partial results are also proved for the category of abelian groups.

On reflexivity and hyperreflexivity of some spaces of intertwining operators

Michal Zajac (2008)

Mathematica Bohemica

Let T , T ' be weak contractions (in the sense of Sz.-Nagy and Foiaş), m , m ' the minimal functions of their C 0 parts and let d be the greatest common inner divisor of m , m ' . It is proved that the space I ( T , T ' ) of all operators intertwining T , T ' is reflexive if and only if the model operator S ( d ) is reflexive. Here S ( d ) means the compression of the unilateral shift onto the space H 2 ⊖ d H 2 . In particular, in finite-dimensional spaces the space I ( T , T ' ) is reflexive if and only if all roots of the greatest common divisor of minimal polynomials...

On strong generation of B(ℋ) by two commutative C*-algebras

R. Berntzen, A. Sołtysiak (1997)

Studia Mathematica

The algebra B(ℋ) of all bounded operators on a Hilbert space ℋ is generated in the strong operator topology by a single one-dimensional projection and a family of commuting unitary operators with cardinality not exceeding dim ℋ. This answers Problem 8 posed by W. Żelazko in [6].

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