This book is essentially a survey of results on the Fuglede-Putnam theorem and its generalizations in a wide variety of directions. Presenting a broad overview of the results obtained in the field since the early 1950s, this is the first monograph to be dedicated to this powerful tool and its variants.
Starting from historical notes and classical versions with their different proofs, the book then explores asymptotic versions, generalizations to non-normal operators, generalizations to unbounded operators, counterexamples, applications, intertwining relations, and conjectures. A rich collection of applications is included.
Aimed at postgraduate students as well as researchers interested in operator theory, this book could also be taught as a specialized course.
Author(s): Mohammed Hichem Mortad
Series: Lecture Notes in Mathematics, 2322
Publisher: Springer
Year: 2022
Language: English
Pages: 163
City: Cham
Preface
Contents
1 Classical Versions and Some Historical Notes
1.1 Finite-Dimensional Versions
1.2 The Classical Fuglede-Putnam Theorem
1.3 Weiss' Theorem Et al.
1.4 Counterexamples
References
2 Generalizations to Bounded Nonnormal Operators
2.1 Barría's Lemma Et al.
2.2 Generalizations Modulo the Hilbert-Schmidt Class
2.3 Generalizations to Subnormal or Hyponormal Operators
2.4 Generalizations to p-Hyponormal or ps: [/EMC pdfmark [/Subtype /Span /ActualText (log) /StPNE pdfmark [/StBMC pdfmarklogps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark-Hyponormal Operators
2.5 A Generalization Based on the Julia Operator
2.6 Further Comments on Other Generalizations
2.7 Counterexamples
References
3 Asymptotic Versions
References
4 Generalizations of the Fuglede-Putnam Theorem to Banach Algebras and Spaces
4.1 Generalizations of the Fuglede-Putnam Theorem to Banach Algebras
4.2 Generalizations of the Fuglede-Putnam Theorem to Banach Spaces
References
5 Generalizations to Unbounded Operators
5.1 All-Unbounded-Operator Versions
5.2 Other Generalizations to Unbounded Operators
5.3 Stochel's Version Involving Unbounded Subnormal and Hyponormal Operators
5.4 Counterexamples
References
6 Some Applications
6.1 Putnam's Applications
6.2 Kaplansky's Theorem
6.3 Devinatz-Nussbaum's Theorem
6.4 Embry's Thereom
6.5 Normality of the Product of Normal Operators
6.6 Normality of the Sum of Normal Operators
6.7 Absolute Values of Products and Sums
6.8 Self-Adjointness of the Normal Product of Self-Adjoint Operators
6.8.1 Commutativity Up to a Factor
6.9 Algebras of Normal Operators
6.10 A Result by F. Kittaneh
6.11 Roots of Normal Operators
6.12 Radjavi-Rosenthal's Theorem
6.13 Certain Maximality Results
6.14 Counterexamples
References
7 Some Other Intertwining Relations
7.1 Beck-Putnam's Theorem and Its Generalizations
7.2 Berberian's Theorem: Generalizations and Applications
7.3 Sheth-Williams' Theorem and Its Generalization
7.4 Operator Equations Involving the Shift Operator
7.5 Counterexamples
References
8 Conjectures
8.1 Conjectures
References
A Bounded and Unbounded Linear Operators
A.1 Bounded Linear Operators
A.2 Unbounded Linear Operators
References
Index
