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On Rank Two Linear Transformations and Reflexivity
Author(s) -
Azoff Edward A.,
Ptak Marek
Publication year - 1996
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/53.2.383
Subject(s) - mathematics , reflexivity , hilbert space , rank (graph theory) , pure mathematics , ideal (ethics) , reflexive operator algebra , operator (biology) , linear map , block (permutation group theory) , operator algebra , order (exchange) , algebra over a field , compact operator , combinatorics , computer science , sociology , epistemology , social science , philosophy , biochemistry , chemistry , repressor , finance , transcription factor , economics , extension (predicate logic) , gene , programming language
We study operator algebras generated by commuting families of nilpotents. In order for such an algebra to be reflexive, it is necessary that each ideal generated by a rank two member of be one‐dimensional. When the underlying space is a finite‐dimensional Hilbert space and the nilpotents in question doubly commute in the sense that they commute with each other's adjoints, the condition is also sufficient. Doubly commuting families of nilpotents admit simultaneous Jordan Canonical Forms and reflexivity of can also be characterized in terms of Jordan block sizes. In particular, our results generalize work of J. Deddens and P. Fillmore on singly‐generated operator algebras.