Dichotomy between operators acting on finite and infinite dimensional Hilbert spaces

Document Type : Research Article

Authors

1 Departamento de Análisis Matem\ático, Facultad de Matemáticas, Instituto de Matemáticas Antonio de Castro Brzezicki, Universidad de Sevilla, Avenida Reina Mercedes, Sevilla, 41080, Spain.

2 Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran.

3 Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Plaza de Ciencias 3, Universidad Complutense de Madrid, Madrid, 28040, Spain.

Abstract

In this expository article, we give several examples showing how drastically different can be the behavior of operators acting on finite versus infinite dimensional Hilbert spaces. This essay is written as in such a friendly-reader to show that the situation in the infinite dimensional setting is trickier than the finite one.

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References

Bart, T. Ehrhardt, and B. Silbermann, Zero sums of idempotents in Banach algebras, Integral Equations Operator Theory 19 (1994), no. 2, 125--134.
Bhatia, Positive definite matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2007.
Eno, On the invariant subspace problem for Banach spaces, Acta Math. 158 (1987), no. 3-4, 213--313.
Eno, On the invariant subspace problem in Hilbert spaces, https://arxiv.org/abs/2305.15442 and https://www.ucm.es/imi/boletin00091#noticia.
K. Fong, H. Radjavi, and P. Rosenthal, Norms for matrices and operators, J. Operator Theory 18 (1987), no. 1, 99--113.
H.-L. Gau, K.-Z. Wang, and P. Y. Wu, Constant norms and numerical radii of matrix powers, Oper. Matrices 13 (2019), no. 4, 1035--1054.
R. Gelbaum and J. M. H. Olmsted, Theorems and counterexamples in mathematics, Problem Books in Mathematics. Springer-Verlag, New York, 1990.
G. Grosse-Erdmann and A. Peris, Linear Chaos, Springer, London, 2011.
Grothendieck, Produits tensoriels topologiques et espaces nucléaires (French), Mem. Amer. Math. Soc. 16 (1955), Chapter 1, 196 pp. Chapter 2, 140 pages.
R. Harris, M. Mazzella, L. J. Patton, D. Renfrew and I. M. Spitkovsky, Numerical ranges of cube roots of the identity, Linear Algebra Appl. 435 (2011), no. 11, 2639--2657.
R. Halmos, A Hilbert Space Problem Book, Second edition. Graduate Texts in Mathematics, 19. Encyclopedia of Mathematics and its Applications, 17. Springer-Verlag, New York-Berlin, 1982.
A. Herrero, Approximation of Hilbert space operators, Vol. 1. Second edition, Pitman Research Notes in Mathematics Series, 224. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.
Herzog, On linear operators having supercyclic vectors, Studia Math. 103 (1992), no. 3, 295--298.
Maroulas, P. J. Psarrakos and M. J. Tsatsomeros, Perron-Frobenius type results on the numerical range, Linear Algebra Appl. 348 (2002) 49--62.
S. Moslehian, Trick with 2✕2 matrices over C*-algebras, Austral. Math. Soc. Gaz. 30 (2003), no. 3, 150--157.
S. Moslehian, K. Sharifi, M. Forough and M. Chakoshi, Moore-Penrose inverses of Gram operators on Hilbert C*-modules, Studia Math. 210 (2012), no. 2, 189--196.
J. Murphy, C*-Algebras and Operator Theory, Academic Press, INC, 1990.
Rolewicz, On orbits of elements, Studia Math. 32 (1969) 17--22.
Simon, Operator Theory, A Comprehensive Course in Analysis, Part 4. American Mathematical Society, Providence, RI, 2015.
Y. Wu, The operator factorization problems, Linear Algebra Appl. 117 (1989) 35--63.
Zhang, Matrix theory. Basic results and techniques, Second edition. Universitext, Springer, New York, 2011.
Journal of the Iranian Mathematical Society
Volume 3, Issue 1
January 2022
Pages 33-41

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