Ranges of Products of Operators

1974 ◽  
Vol 26 (6) ◽  
pp. 1430-1441 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Banach Space ◽  
Dual Problem ◽  
Bounded Operator ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Operators

Suppose that T and A are bounded linear operators. In this paper we examine the relation between the ranges of A and TA, under various additional hypotheses on T and A. We also consider the dual problem of the relation between the null-spaces of T and AT; and we consider some cases where T or A are only closed operators. Our major results about ranges of bounded operators are summarized in the following theorem.Theorem 1. Suppose that T is a bounded operator on a Banach space E and that A is a non-zero bounded operator from some Banach space to E.

Descent spectrum equality

2019 ◽  
Vol 12 (02) ◽  
pp. 1950029
Author(s):  
Abdelaziz Tajmouati ◽  
Hamid Boua ◽  
Abdeslam El Bakkali
Keyword(s):  
Banach Space ◽  
Bounded Operator ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Single Operator

A bounded operator [Formula: see text] in a Banach space [Formula: see text] is said to satisfy the descent spectrum equality, if the descent spectrum of [Formula: see text] as an operator on [Formula: see text] coincides with the descent spectrum of [Formula: see text] as an element of the algebra [Formula: see text] of all bounded linear operators on [Formula: see text]. In this paper, we give some conditions under which the equality [Formula: see text] holds for a single operator [Formula: see text].

Strictly Singular and Cosingular Multiplications

2005 ◽  
Vol 57 (6) ◽  
pp. 1249-1278 ◽  
Author(s):  
Mikael Lindström ◽  
Eero Saksman ◽  
Hans-Olav Tylli
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Strictly Singular ◽  
Classical Banach Space

AbstractLet L(X) be the space of bounded linear operators on the Banach space X. We study the strict singularity and cosingularity of the two-sidedmultiplication operators S ↦ ASB on L(X), where A, B ∈ L(X) are fixed bounded operators and X is a classical Banach space. Let 1 < p < ∞and p ≠ 2. Our main result establishes that the multiplication S ↦ ASB is strictly singular on L(Lp(0, 1)) if and only if the non-zero operators A, B ∈ L(Lp(0, 1)) are strictly singular. We also discuss the case where X is a L1- or a L∞-space, as well as several other relevant examples.

scholarly journals Uniqueness of the maximal ideal of operators on the ℓp-sum of ℓ∞n (n ∈ ) for 1 < p < ∞

2016 ◽  
Vol 160 (3) ◽  
pp. 413-421 ◽  
Author(s):  
TOMASZ KANIA ◽  
NIELS JAKOB LAUSTSEN
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
American Mathematical Society ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Unique Maximal Ideal

AbstractA recent result of Leung (Proceedings of the American Mathematical Society, 2015) states that the Banach algebra ℬ(X) of bounded, linear operators on the Banach space X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓ1 contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓp and X = (⊕n∈$\mathbb{N}$ ℓ1n)ℓp whenever p ∈ (1, ∞).

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

scholarly journals On the n-parameter abstract Cauchy problem

2004 ◽  
Vol 69 (3) ◽  
pp. 383-394
Author(s):  
M. Janfada ◽  
A. Niknam
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Unique Solution ◽  
Initial Value Problems ◽  
Abstract Cauchy Problem ◽  
Linear Operators ◽  
Uniqueness Of Solution ◽  
Initial Value ◽  
Bounded Linear ◽  
Closed Operators

Let Hi(i = 1, 2, …, n), be closed operators in a Banach space X. The generalised initialvalue problem of the abstract Cauchy problem is studied. We show that the uniqueness of solution u: [0, T1] × [0, T2] × … × [0, Tn] → X of this n-abstract Cauchy problem is closely related to C0-n-parameter semigroups of bounded linear operators on X. Also as another application of C0-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

1969 ◽  
Vol 21 ◽  
pp. 592-594 ◽  
Author(s):  
A. F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Finite Number ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

On the invertibility of length two elementary operators

2015 ◽  
Vol 30 ◽  
pp. 916-913
Author(s):  
Janko Bracic ◽  
Nadia Boudi
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Elementary Operator ◽  
Bounded Linear Operators ◽  
Existence Of A Solution ◽  
Bounded Linear ◽  
Elementary Operators ◽  
Necessary And Sufficient

Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.

Common spectral properties of linear operators A and B satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

2019 ◽  
Vol 12 (05) ◽  
pp. 1950084
Author(s):  
Anuradha Gupta ◽  
Ankit Kumar
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Spectral Properties ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

Let [Formula: see text] and [Formula: see text] be two bounded linear operators on a Banach space [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] and [Formula: see text], then [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] have some common spectral properties. Drazin invertibility and polaroidness of these operators are also discussed. Cline’s formula for Drazin inverse in a ring with identity is also studied under the assumption that [Formula: see text] for some positive integer [Formula: see text].

scholarly journals GROWTH CONDITIONS FOR OPERATORS WITH SMALLEST SPECTRUM

2014 ◽  
Vol 57 (3) ◽  
pp. 665-680
Author(s):  
H. S. MUSTAFAYEV
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Growth Conditions ◽  
Invertible Operator ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Formula Group ◽  
Image Position

AbstractLet A be an invertible operator on a complex Banach space X. For a given α ≥ 0, we define the class $\mathcal{D}$Aα(ℤ) (resp. $\mathcal{D}$Aα (ℤ+)) of all bounded linear operators T on X for which there exists a constant CT>0, such that $ \begin{equation*} \Vert A^{n}TA^{-n}\Vert \leq C_{T}\left( 1+\left\vert n\right\vert \right) ^{\alpha }, \end{equation*} $ for all n ∈ ℤ (resp. n∈ ℤ+). We present a complete description of the class $\mathcal{D}$Aα (ℤ) in the case when the spectrum of A is real or is a singleton. If T ∈ $\mathcal{D}$A(ℤ) (=$\mathcal{D}$A0(ℤ)), some estimates for the norm of AT-TA are obtained. Some results for the class $\mathcal{D}$Aα (ℤ+) are also given.

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