# Transitivity for linear operators on a Banach space

Studia Mathematica (1999)

• Volume: 132, Issue: 3, page 239-243
• ISSN: 0039-3223

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## Abstract

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Let G be the multiplicative group of invertible elements of E(X), the algebra of all bounded linear operators on a Banach space X. In 1945 Mackey showed that if ${x}_{1},\dots ,{x}_{n}$ and ${y}_{1},\dots ,{y}_{n}$ are any two sets of linearly independent elements of X with the same number of items, then there exists T ∈ G so that $T\left({x}_{k}\right)={y}_{k}$, $k=1,\dots ,n$. We prove that some proper multiplicative subgroups of G have this property.

## How to cite

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Yood, Bertram. "Transitivity for linear operators on a Banach space." Studia Mathematica 132.3 (1999): 239-243. <http://eudml.org/doc/216597>.

@article{Yood1999,
abstract = {Let G be the multiplicative group of invertible elements of E(X), the algebra of all bounded linear operators on a Banach space X. In 1945 Mackey showed that if $x_1,…,x_n$ and $y_1,…,y_n$ are any two sets of linearly independent elements of X with the same number of items, then there exists T ∈ G so that $T(x_k) = y_k$, $k = 1,…,n$. We prove that some proper multiplicative subgroups of G have this property.},
author = {Yood, Bertram},
journal = {Studia Mathematica},
keywords = {multiplicative subgroup of invertible elements; linearly independent transitive subset},
language = {eng},
number = {3},
pages = {239-243},
title = {Transitivity for linear operators on a Banach space},
url = {http://eudml.org/doc/216597},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Yood, Bertram
TI - Transitivity for linear operators on a Banach space
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 3
SP - 239
EP - 243
AB - Let G be the multiplicative group of invertible elements of E(X), the algebra of all bounded linear operators on a Banach space X. In 1945 Mackey showed that if $x_1,…,x_n$ and $y_1,…,y_n$ are any two sets of linearly independent elements of X with the same number of items, then there exists T ∈ G so that $T(x_k) = y_k$, $k = 1,…,n$. We prove that some proper multiplicative subgroups of G have this property.
LA - eng
KW - multiplicative subgroup of invertible elements; linearly independent transitive subset
UR - http://eudml.org/doc/216597
ER -

## References

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1. [1] S. Banach, Théorie des opérations linéaires, Warszawa, 1932. Zbl0005.20901
2. [2] S. R. Caradus, W. E. Pfaffenberger and B. Yood, Calkin Algebras and Algebras of Operators on Banach Spaces, Marcel Dekker, New York, 1974. Zbl0299.46062
3. [3] P. Civin and B. Yood, Involutions on Banach algebras, Pacific J. Math. 9 (1959), 415-436. Zbl0086.09601
4. [4] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York, 1965.
5. [5] G. W. Mackey, On infinite-dimensional linear spaces, Trans. Amer. Math. Soc. 57 (1945), 155-207. Zbl0061.24301
6. [6] B. Yood, Transformations between Banach spaces in the uniform topology, Ann. of Math. 50 (1949), 486-503. Zbl0034.06401

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