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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 operators Cauchy dual to 2-hyperexpansive operators: the unbounded case

Sameer Chavan (2011)

Studia Mathematica

The Cauchy dual operator T’, given by T ( T * T ) - 1 , provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1 and 1.2 below)....

On operators close to isometries

Sameer Chavan (2008)

Studia Mathematica

We introduce and discuss a class of operators, to be referred to as operators close to isometries. The Bergman-type operators, 2-hyperexpansions, expansive p-isometries, and certain alternating hyperexpansions are main examples of such operators. We establish a few decomposition theorems for operators close to isometries. Applications are given to the theory of p-isometries and of hyperexpansive operators.

On the Range and the Kernel of Derivations

Bouali, Said, Bouhafsi, Youssef (2006)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 47B47, 47B10; Secondary 47A30.Let H be a separable infinite dimensional complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A ∈ L(H), the derivation δA : L(H)→ L(H) is defined by δA(X) = AX-XA. In this paper we prove that if A is an n-multicyclic hyponormal operator and T is hyponormal such that AT = TA, then || δA(X)+T|| ≥ ||T|| for all X ∈ L(H). We establish the same inequality if A is...

On the spectral multiplicity of a direct sum of operators

M. T. Karaev (2006)

Colloquium Mathematicae

We calculate the spectral multiplicity of the direct sum T⊕ A of a weighted shift operator T on a Banach space Y which is continuously embedded in l p and a suitable bounded linear operator A on a Banach space X.

On weak supercyclicity II

Carlos S. Kubrusly, Bhagwati P. Duggal (2018)

Czechoslovak Mathematical Journal

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly l -sequentially supercyclic, and (iii) weak l -sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space operators:...

Operators with hypercyclic Cesaro means

Fernando León-Saavedra (2002)

Studia Mathematica

An operator T on a Banach space ℬ is said to be hypercyclic if there exists a vector x such that the orbit T x n 1 is dense in ℬ. Hypercyclicity is a strong kind of cyclicity which requires that the linear span of the orbit is dense in ℬ. If the arithmetic means of the orbit of x are dense in ℬ then the operator T is said to be Cesàro-hypercyclic. Apparently Cesàro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesàro-hypercyclic if and only if there exists a vector...

Orbits of linear operators and Banach space geometry

Jean-Matthieu Augé (2012)

Studia Mathematica

Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents q > 0 such that for every non-nilpotent operator T, there exists x ∈ X such that ( | | T x | | / | | T | | ) q ( ) , using techniques which involve the modulus of asymptotic uniform smoothness of X.

Orbits under a class of isometries of L¹[0,1]

Terje Hõim (2004)

Studia Mathematica

We study the orbits of isometries of L¹[0,1]. For a certain class of isometries we show that the set of functions f in L¹[0,1] for which the orbit of f under the isometry T is equivalent to the usual canonical basis e₁,e₂,e₃,... of l¹ is an open dense set. In the proof we develop a new method to get copies of l¹ inside L¹[0,1] using geometric progressions. This method does not use disjoint or relatively disjoint supports, which seems to be the most common way to get such copies. We also use this...

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