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1 - Linear Operators and Matrices
from Part I - Numerical Linear Algebra
Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
Book:

Classical Numerical Analysis

Published online:

29 September 2022

Print publication:

20 October 2022, pp 3-19
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Summary

We present several facts about the natural transformations between vector spaces, and their representations via matrices. We introduce induced matrix norms, and the spectral decomposition of nondefective matrices

1 - Semigroups and Generators
David Applebaum, University of Sheffield
Book:

Semigroups of Linear Operators

Published online:

27 July 2019

Print publication:

15 August 2019, pp 9-30
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Summary

We define semigroups of linear operators in Banach spaces and introduce their generators (which may be unbounded operators) and their resolvents. A number of examples are given, which will be referred back to throughout the volume.

Value Distribution and Linear Operators
R. Halburd, R. Korhonen
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 57 / Issue 2 / June 2014

Published online by Cambridge University Press:

22 November 2013, pp. 493-504
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Nevanlinna's second main theorem is a far-reaching generalization of Picard's theorem concerning the value distribution of an arbitrary meromorphic function f. The theorem takes the form of an inequality containing a ramification term in which the zeros and poles of the derivative f′ appear. We show that a similar result holds for special subfields of meromorphic functions where the derivative is replaced by a more general linear operator, such as higher-order differential operators and differential-difference operators. We subsequently derive generalizations of Picard's theorem and the defect relations.

Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices
Wai-Shun Cheung, Chi-Kwong Li
Journal:

Canadian Mathematical Bulletin / Volume 44 / Issue 3 / 01 September 2001

Published online by Cambridge University Press:

20 November 2018, pp. 270-281

Print publication:

01 September 2001
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Let $c\,=\,\left( {{c}_{1}},\ldots ,{{c}_{n}} \right)$ be such that ${{c}_{1}}\,\ge \,\cdots \,\ge \,{{c}_{n}}$. The $c$-numerical range of an $n\,\times \,n$ matrix $A$ is defined by
$${{W}_{c}}\left( A \right)\,=\,\left\{ \sum\limits_{j=1}^{n}{{{c}_{j}}\left( A{{x}_{j}},\,{{x}_{j}} \right)\,:\,\left\{ {{x}_{1}},\ldots ,{{x}_{n}} \right\}\,\text{an}\,\text{orthonormal basis for }{{\mathbf{C}}^{n}}} \right\}\,,$$
and the $c$-numerical radius of $A$ is defined by ${{r}_{c}}\left( A \right)\,=\,\max \left\{ \left| z \right|\,:\,z\,\in \,{{W}_{c}}\left( A \right) \right\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying
$${{W}_{c}}\left( \phi \left( A \right) \right)={{W}_{c}}\left( A \right)\text{for}\,\,\text{all}\,\,A\,\text{or}\,\,{{r}_{c}}\left( \phi \left( A \right) \right)={{r}_{c}}\left( A \right)\text{for}\,\,\text{all}\,A.$$

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