Linear Operators on a Banach Space

2001 ◽  
pp. 211-235
Author(s):  
Israel Gohberg ◽  
Seymour Goldberg
Keyword(s):  
Banach Space ◽  
Linear Operators

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

scholarly journals The local spectral radius of a nonnegative orbit of compact linear operators

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Mihály Pituk
Keyword(s):  
Banach Space ◽  
Spectral Radius ◽  
Additional Assumption ◽  
Real Banach Space ◽  
Linear Operators ◽  
Partial Ordering ◽  
Positive Eigenvector ◽  
Local Spectral Radius

AbstractWe consider orbits of compact linear operators in a real Banach space which are nonnegative with respect to the partial ordering induced by a given cone. The main result shows that under a mild additional assumption the local spectral radius of a nonnegative orbit is an eigenvalue of the operator with a positive eigenvector.

An operator version of Abel's continuity theorem

2010 ◽  
Vol 17 (4) ◽  
pp. 787-794
Author(s):  
Vaja Tarieladze
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Identity Operator ◽  
Continuous Linear ◽  
Continuity Theorem ◽  
Operator Version ◽  
Continuous Linear Operators

Abstract For a Banach space X let 𝔄 be the set of continuous linear operators A : X → X with ‖A‖ < 1, I be the identity operator and 𝔄 c ≔ {A ∈ 𝔄 : ‖I – A‖ ≤ c(1 – ‖A‖)}, where c ≥ 1 is a constant. Let, moreover, (xk ) k≥0 be a sequence in X such that the series converges and ƒ : 𝔄 ∪ {I} → X be the mapping defined by the equality It is shown that ƒ is continuous on 𝔄 and for every c ≥ 1 the restriction of ƒ to 𝔄 c ∪ {I} is continuous at I.

scholarly journals THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE

2006 ◽  
Vol 49 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Yun Sung Choi ◽  
Domingo Garcia ◽  
Sung Guen Kim ◽  
Manuel Maestre
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Banach Spaces ◽  
Compact Space ◽  
Linear Operators ◽  
Homogeneous Polynomials ◽  
Numerical Radius ◽  
Disc Algebra ◽  
Multilinear Maps ◽  
Numerical Index

AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

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 The numerical ranges of unbounded linear operators

1975 ◽  
Vol 12 (1) ◽  
pp. 23-25 ◽  
Author(s):  
Béla Bollobás ◽  
Stephan E. Eldridge
Keyword(s):  
Banach Space ◽  
Scalar Field ◽  
Banach Spaces ◽  
Unbounded Operator ◽  
Numerical Range ◽  
Linear Operators ◽  
Density Property ◽  
Unbounded Linear Operators

Giles and Joseph (Bull. Austral. Math. Soc. 11 (1974), 31–36), proved that the numerical range of an unbounded operator on a Banach space has a certain density property. They showed, in particular, that the numerical range of an unbounded operator on certain Banach spaces is dense in the scalar field. We prove that the numerical range of an unbounded operator on a Banach space is always dense in the scalar field.

scholarly journals Some properties of shift operators on algebras generated by $*$-polynomials

2018 ◽  
Vol 10 (1) ◽  
pp. 206-212
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Holomorphic Functions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Countable System ◽  
Shift Operators ◽  
Underlying Space ◽  
Algebra Of Functions ◽  
Algebras Of Holomorphic Functions ◽  
Proper Subalgebra

A $*$-polynomial is a function on a complex Banach space $X,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for non-negative integers $p$ and $q,$ a $(p,q)$-polynomial is a function on $X,$ which is the restriction to the diagonal of some mapping, defined on the Cartesian power $X^{p+q},$ which is linear with respect to every of its first $p$ arguments and antilinear with respect to every of its other $q$ arguments. The set of all continuous $*$-polynomials on $X$ form an algebra, which contains the algebra of all continuous polynomials on $X$ as a proper subalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions than algebras of holomorphic functions. On the other hand, due to the similarity of structures of $*$-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces. We investigate the Frechet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous $*$-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.

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.

Continuity Properties of Operator Spectra

1977 ◽  
Vol 29 (2) ◽  
pp. 429-437 ◽  
Author(s):  
Nicholas J. Bezak ◽  
Martin Eisen
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Continuity Properties

This paper is devoted to the study of convergence and variation of operator spectra with respect to the distance G of Gokhburg and Markus [5] for subspaces and linear operators in a Banach space. We use the convention of Kato [7] and refer to convergence with respect to G as generalizedconvergence.

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