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).

scholarly journals Asymptotically quasi-compact products of bounded linear operators

1973 ◽  
Vol 16 (3) ◽  
pp. 286-289 ◽  
Author(s):  
Anthony F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Present Note ◽  
Fredholm Determinant ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

It is known (see, for instance, [1] p. 64, [6] p. 264) that, if A and B are bounded linear operators on a Banach space into itself (or, more generally, if A is a bounded linear operator on into a Banach space and B is a bounded linear operator on into), then AB and BA have the same spectrum except (possibly) for zero. In the present note, it is shown that AB is asymptotically quasi-compact if and only if BA is asymptotically quasi-compact, and that then any Fredholm determinant for AB is a Fredholm determinant for BA and vice versa.

On Certain Classes of Bounded Linear Operators

1970 ◽  
Vol 13 (4) ◽  
pp. 469-473
Author(s):  
C-S Lin
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Fredholm Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear

Let T—c be a Fredholm operator, where T is a bounded linear operator on a complex Banach space and c is a scalar, the set of all such scalars is called the Φ-set of T [2] and was studied by many authors. In this connection, the purpose of the present paper is to investigate some classes Φ(V) of all such operators for any subset V of the complex plane.Let X be a Banach space over the field C of complex numbers with dim Z = ∞, unless otherwise stated, B(X) the Banach algebra of all bounded linear operators and K(X) the closed two-sided ideal of all compact operators on X.

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 A direct proof of a theorem of West on sequences of Riesz operators

1974 ◽  
Vol 15 (2) ◽  
pp. 93-94
Author(s):  
Anthony F. Ruston
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Direct Proof ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Riesz Operator ◽  
Bounded Linear ◽  
Riesz Operators

We recall (cf. [2] Definitions 3.1 and 3.2, p. 322) that a bounded linear operator T on a Banach space ℵ into itself is said to be asymptotically quasi-compact if K(Tn)⅟n → 0 as n → ∞. where K(U) = inf ∥U–C∥ for every bounded linear operator U on ℵ into itself, the infimum being taken over all compact linear operators C on ℵ into itself. For a complex Banach space, this is equivalent (cf. [2], pp. 319, 321 and 326) to T being a Riesz operator.

scholarly journals Support projections on Banach spaces

1977 ◽  
Vol 18 (1) ◽  
pp. 13-15 ◽  
Author(s):  
P. G. Spain
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Support Projection

Each bounded linear operator a on a Hilbert space K has a hermitian left-support projection p such that and (1 – p)K = ker α* = ker αα*. I demonstrate here that certain operators on Banach spaces also have left supports.Throughout this paper X will be a complex Banach space with norm-dual X', and L(X) will be the Banach algebra of bounded linear operators on X. Two linear subspaces Y and Z of X are orthogonal (in the sense of G. Birkhoff) if ∥ y ∥ ≦ ∥ y + z ∥ (y ∈Y, z ∈ Z); this orthogonality relation is not, in general, symmetric. It is easy to see that pX is orthogonal to (1 – p)X if and only if the norm of p is 0 or 1, when p is a projection on X.

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.

scholarly journals Expansion of analytic functions of an operator in series of Faber polynomials

1997 ◽  
Vol 56 (2) ◽  
pp. 303-318 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Analytic Functions ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Faber Polynomials ◽  
Bounded Linear ◽  
In Series ◽  
Image Position

Let T: B → B be a bounded linear operator on the complex Banach space B and let f(z) be analytic on a domain D containing the spectrum Sp(T) of T. Then f(T) is defined bywhere C is a contour surrounding SP(T) and contained in D.

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.

scholarly journals On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Aftab Khan ◽  
Gul Rahmat ◽  
Akbar Zada
Keyword(s):  
Banach Space ◽  
Exponential Stability ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Uniform Exponential Stability ◽  
Discrete Semigroup ◽  
Exponentially Stable ◽  
Discrete Semigroups

We prove that a discrete semigroup𝕋={T(n):n∈ℤ+}of bounded linear operators acting on a complex Banach spaceXis uniformly exponentially stable if and only if, for eachx∈AP0(ℤ+,X), the sequencen↦∑k=0n‍T(n-k)x(k):ℤ+→Xbelongs toAP0(ℤ+,X). Similar results for periodic discrete evolution families are also stated.

scholarly journals Joint spectra of operators on Banach space

1986 ◽  
Vol 28 (1) ◽  
pp. 69-72 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Point Spectrum ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Joint Spectrum ◽  
Approximate Point Spectrum ◽  
Definition Of ◽  
Unit Vectors

Let X be a complex Banach space. We denote by B(X) the algebra of all bounded linear operators on X. Let = (T1, …, Tn) be a commuting n-tuple of operators on X. And let στ() and σ″() by Taylor's joint spectrum and the doubly commutant spectrum of , respectively. We refer the reader to Taylor [8] for the definition of στ() and σ″(), A point z = (z1,…, zn) of ℂn is in the joint approximate point spectrum σπ() of if there exists a sequence {xk} of unit vectors in X such that∥(Ti – zi)xk∥→0 as k → ∞ for i = 1, 2,…, n.

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