Dilations, Completely Positive Maps and Geometry
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Beschreibung
This book introduces the dilation theory of operators on Hilbert spaces and its relationship to complex geometry. Classical as well as very modern topics are covered in the book. On the one hand, it introduces the reader to the characteristic function, a classical object used by Sz.-Nagy and Foias and still a topic of current research. On the other hand, it describes the dilation theory of the symmetrized bidisc which has been developed mostly in the present century and is a very active topic of research. It also describes an abstract theory of dilation in the setting of set theory. This was developed very recently.
A good portion of the book discusses various geometrical objects like the bidisc, the Euclidean unit ball, and the symmetrized bidisc. It shows the similarities and differences between the dilation theory in these domains. While completely positive maps play a big role in the dilation theory of the Euclidean unit ball, this is not so in the symmetrized bidisc for example. There, the central role is played by an operator equation. Targeted to graduate students and researchers, the book introduces the reader to different techniques applicable in different domains.
This book introduces the dilation theory of operators on Hilbert spaces and its relationship to complex geometry. Classical as well as very modern topics are covered in the book. On the one hand, it introduces the reader to the characteristic function, a classical object used by Sz.-Nagy and Foias and still a topic of current research. On the other hand, it describes the dilation theory of the symmetrized bidisc which has been developed mostly in the present century and is a very active topic of research. It also describes an abstract theory of dilation in the setting of set theory. This was developed very recently.
A good portion of the book discusses various geometrical objects like the bidisc, the Euclidean unit ball, and the symmetrized bidisc. It shows the similarities and differences between the dilation theory in these domains. While completely positive maps play a big role in the dilation theory of the Euclidean unit ball, this is not so in the symmetrized bidisc for example. There, the central role is played by an operator equation. Targeted to graduate students and researchers, the book introduces the reader to different techniques applicable in different domains.
Covers classical as well as very modern topics in the dilation theory Deals with the dilation theory of operators on Hilbert spaces and its relationship to complex geometry Introduces to the characteristic function, a classical object used by Sz.-Nagy and Foias
Autor*in
B.V. Rajarama Bhat
Themen in »Dilations, Completely Positive Maps and Geometry«
operator theory self-adjoint operator algebras reproducing kernel symmetrized bi-disc functional analysis general theory of operators spectral sets of linear operators Hilbert spaces model theory spectral sets compressions of operators extensions of of operators dilations of operators
Stimmen zu »Dilations, Completely Positive Maps and Geometry«
“In this book, the authors obtain concrete constructions of dilations of commuting operator tuples related to four domains where this theory has been beautifully successful so far. … The constructions in this book are very interesting because of various inputs that are required, like, for example, the Fejér-Riesz theorem. So, it connects to holomorphic function theory and geometry very well.” (Luo Yi Shi, Mathematical Reviews, February, 2025)
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