scholarly journals What is... a Hereditarily Indecomposable Banach Space?

2019 ◽  
Vol 66 (09) ◽  
pp. 1
Author(s):  
Hossein Hosseini Giv
Keyword(s):  
Banach Space ◽  
Hereditarily Indecomposable Banach Space ◽  
Indecomposable Banach Space ◽  
Hereditarily Indecomposable

scholarly journals Quotient Hereditarily Indecomposable Banach Spaces

1999 ◽  
Vol 51 (3) ◽  
pp. 566-584 ◽  
Author(s):  
V. Ferenczi
Keyword(s):  
Banach Space ◽  
Singular Perturbation ◽  
Banach Spaces ◽  
Infinite Dimensional ◽  
Strictly Singular ◽  
Hereditarily Indecomposable Banach Spaces ◽  
Hereditarily Indecomposable

AbstractA Banach space X is said to be quotient hereditarily indecomposable if no infinite dimensional quotient of a subspace of X is decomposable. We provide an example of a quotient hereditarily indecomposable space, namely the space XGM constructed by W. T. Gowers and B. Maurey in [GM]. Then we provide an example of a reflexive hereditarily indecomposable space whose dual is not hereditarily indecomposable; so is not quotient hereditarily indecomposable. We also show that every operator on * is a strictly singular perturbation of an homothetic map.

scholarly journals MAXIMAL IDEALS IN THE ALGEBRA OF OPERATORS ON CERTAIN BANACH SPACES

2002 ◽  
Vol 45 (3) ◽  
pp. 523-546 ◽  
Author(s):  
Niels Jakob Laustsen
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
Operator Ideal ◽  
Linear Operators ◽  
Weakly Compact ◽  
Maximal Ideals ◽  
Hereditarily Indecomposable Banach Space ◽  
Unique Maximal Ideal ◽  
The Ideal

AbstractFor a Banach space $\mathfrak{X}$, let $\mathcal{B}(\mathfrak{X})$ denote the Banach algebra of all continuous linear operators on $\mathfrak{X}$. First, we study the lattice of closed ideals in $\mathcal{B}(\mathfrak{J}_p)$, where $1 \lt p \t \infty$ and $\mathfrak{J}_p$ is the $p$th James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{J}_p)$. Applications of this result include the following.(i) The Brown–McCoy radical of $\mathcal{B}(\mathfrak{X})$, which by definition is the intersection of all maximal ideals in $\mathcal{B}(\mathfrak{X})$, cannot be turned into an operator ideal. This implies that there is no ‘Brown–McCoy’ analogue of Pietsch’s construction of the operator ideal of inessential operators from the Jacobson radical of $\mathcal{B}(\mathfrak{X})/\mathcal{A}(\mathfrak{X})$.(ii) For each natural number $n$ and each $n$-tuple $(m_1,\dots,m_n)$ in $\{k^2\mid k\in\mathbb{N}\}\cup\{\infty\}$, there is a Banach space $\mathfrak{X}$ such that $\mathcal{B}(\mathfrak{X})$ has exactly $n$ maximal ideals, and these maximal ideals have codimensions $m_1,\dots,m_n$ in $\mathcal{B}(\mathfrak{X})$, respectively; the Banach space $\mathfrak{X}$ is a finite direct sum of James spaces and $\ell_p$-spaces.Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces $\mathfrak{X}$ such that all the maximal ideals in $\mathcal{B}(\mathfrak{X})$ can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{X})$ for each hereditarily indecomposable Banach space $\mathfrak{X}$, and we prove that there are $2^{2^{\aleph_0}}$ distinct maximal ideals in $\mathcal{B}(\mathfrak{G})$, where $\mathfrak{G}$ is the Banach space constructed by Gowers to solve Banach’s hyperplane problem.AMS 2000 Mathematics subject classification: Primary 47D30; 47D50; 46H10; 16D30

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