Quasi-compactness of linear operators on Banach spaces: New properties and application to Markov chains

2020 ◽  
pp. 2150060
Author(s):  
Leila Mebarki ◽  
Bekkai Messirdi ◽  
Mohammed Benharrat
Keyword(s):  
Banach Space ◽  
Markov Chains ◽  
Banach Spaces ◽  
Invariant Subspaces ◽  
Linear Operators ◽  
Fredholm Alternative ◽  
Riesz Operators ◽  
Algebraic Operations ◽  
Mathematical Literature ◽  
Schauder Theorem

The purpose of this paper is to study the notion of quasi-compact linear operators acting in a Banach space. This class of operators contains the set of compact, polynomially compact, quasi-nilpotent and that of all Riesz operators. We show the equivalence between different definitions of quasi-compactness known in the mathematical literature and we present several general theorems about quasi-compact endomorphisms: stability under algebraic operations, extension of Schauder theorem and the Fredholm alternative. We also study the question of existence of invariant subspaces and we examine the class of semigroups for quasi-compact operators. The obtained results are used to describe Markov chains.

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 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 On Riesz operators

1969 ◽  
Vol 16 (3) ◽  
pp. 227-232 ◽  
Author(s):  
J. C. Alexander
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Operators ◽  
Image Position ◽  
Schauder’S Theorem

In (4) Vala proves a generalization of Schauder's theorem (3) on the compactness of the adjoint of a compact linear operator. The particular case of Vala's result that we shall be concerned with is as follows. Let t1 and t2 be non-zero bounded linear operators on the Banach spaces Y and X respectively, and denote by 1T2 the operator on B(X, Y) defined by

scholarly journals ON A CHARACTERISTIC PROPERTY OF FINITE-DIMENSIONAL BANACH SPACES

2011 ◽  
Vol 53 (3) ◽  
pp. 443-449 ◽  
Author(s):  
ANTONÍN SLAVÍK
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Characteristic Property ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Counter Example ◽  
Finite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Dvoretzky’S Theorem ◽  
The Given

AbstractThis paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

scholarly journals Spectral integration and spectral theory for non-Archimedean Banach spaces

2002 ◽  
Vol 31 (7) ◽  
pp. 421-442 ◽  
Author(s):  
S. Ludkovsky ◽  
B. Diarra
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Algebras ◽  
Linear Operators ◽  
Spectral Integration ◽  
Different Types ◽  
Commutative Subalgebras ◽  
Continuous Operators ◽  
Stone Theorem ◽  
Continuous Linear Operators

Banach algebras over arbitrary complete non-Archimedean fields are considered such that operators may be nonanalytic. There are different types of Banach spaces over non-Archimedean fields. We have determined the spectrum of some closed commutative subalgebras of the Banach algebraℒ(E)of the continuous linear operators on a free Banach spaceEgenerated by projectors. We investigate the spectral integration of non-Archimedean Banach algebras. We define a spectral measure and prove several properties. We prove the non-Archimedean analog of Stone theorem. It also contains the case ofC-algebrasC∞(X,𝕂). We prove a particular case of a representation of aC-algebra with the help of aL(Aˆ,μ,𝕂)-projection-valued measure. We consider spectral theorems for operators and families of commuting linear continuous operators on the non-Archimedean Banach space.

On Spaces of Compact Operators in Non-Archimedean Banach Spaces

1989 ◽  
Vol 32 (4) ◽  
pp. 450-458
Author(s):  
Takemitsu Kiyosawa
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convergent Sequence ◽  
Compact Operators ◽  
Linear Operators ◽  
Continuous Linear ◽  
Infinite Dimensional ◽  
Valued Field ◽  
Infinite Dimensional Banach Space ◽  
Continuous Linear Operators

AbstractLet K be a non-trivial complete non-Archimedean valued field and let E be an infinite-dimensional Banach space over K. Some of the main results are:(1) K is spherically complete if and only if every weakly convergent sequence in l∞ is norm-convergent.(2) If the valuation of K is dense, then C0 is complemented in E if and only if C(E,c0) is n o t complemented in L(E,c0), where L(E,c0) is the space of all continuous linear operators from E to c0 and C(E,c0) is the subspace of L(E, c0) consisting of all compact linear operators.

LIE-TYPE DERIVATIONS OF NEST ALGEBRAS ON BANACH SPACES AND RELATED TOPICS

2021 ◽  
pp. 1-40
Author(s):  
FENG WEI ◽  
YUHAO ZHANG
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Mapping ◽  
Linear Operators ◽  
Future Research ◽  
Complex Field ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Nest Algebras

Abstract Let $\mathcal {X}$ be a Banach space over the complex field $\mathbb {C}$ and $\mathcal {B(X)}$ be the algebra of all bounded linear operators on $\mathcal {X}$ . Let $\mathcal {N}$ be a nontrivial nest on $\mathcal {X}$ , $\text {Alg}\mathcal {N}$ be the nest algebra associated with $\mathcal {N}$ , and $L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\ldots ,x_n)$ is an $(n-1)\,$ th commutator defined by n indeterminates $x_1, x_2, \ldots , x_n$ . It is shown that L satisfies the rule $$ \begin{align*}L(p_n(A_1, A_2, \ldots, A_n))=\sum_{k=1}^{n}p_n(A_1, \ldots, A_{k-1}, L(A_k), A_{k+1}, \ldots, A_n) \end{align*} $$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ if and only if there exist a linear derivation $D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ and a linear mapping $H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$ vanishing on each $(n-1)\,$ th commutator $p_n(A_1,A_2,\ldots , A_n)$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \text {Alg}\mathcal {N}$ . We also propose some related topics for future research.

ON THE DECOMPOSITION OF OPERATORS WITH SEVERAL ALMOST-INVARIANT SUBSPACES

2019 ◽  
Vol 99 (2) ◽  
pp. 274-283
Author(s):  
AMANOLLAH ASSADI ◽  
MOHAMAD ALI FARZANEH ◽  
HAJI MOHAMMAD MOHAMMADINEJAD
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Closed Subspace ◽  
Invariant Subspaces ◽  
Weak Limit ◽  
Linear Operators ◽  
Sufficient Condition ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Rank Operator

We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

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