Lebesgue Decomposition for Positive Operators Revisited

C. R. Math. Rep. Acad. Sci. Canada Vol. 45 (2) 2023, pp. 37–55
(Received: 2023-06-29 )

Yoshiki Aibara, Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan; e-mail: y.aibara.math95@gmail.com

Yoshimichi Ueda, Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan; e-mail: ueda@math.nagoya-u.ac.jp

Abstract/Résumé:

We explain how Pusz–Woronowicz’s notion of functional calculus fits the theory of Lebesgue decomposition for positive operators on Hilbert spaces initially developed by Ando. In this way, we reconstruct the essential and fundamental part of the theory.

On montre comment la notion de calcul fonctionnel de Pusz–Woronomicz s’adapte à la théorie de décomposition de Lebesgue pour les opérateurs positifs sur un espace de Hilbert, initialement dévelopée par Ando. De cette façon on reconstruit la partie essentielle et fondamentale de cette théorie.

Keywords: Binary operation, Functional calculus, Lebesgue decomposition, Positive operator

AMS Subject Classification: None of the above; but in this section, Functional calculus, Operator means; shorted operators; etc., 28E99, 47A60, 47A64, 47B02

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