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Definitizability of Certain Functions and the Existence of Eigenvectors of Unitary Operators in Pontrjagin Spaces
Author(s) -
Sasvári Zoltán
Publication year - 1988
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19881350106
Subject(s) - mathematics , unitary state , eigenvalues and eigenvectors , pure mathematics , space (punctuation) , polynomial , commutative property , section (typography) , operator (biology) , combinatorics , mathematical analysis , linguistics , philosophy , physics , biochemistry , chemistry , repressor , quantum mechanics , political science , transcription factor , law , gene , advertising , business
Let P k c ( G ) denote the set of continuous functions with k negative squares on a locally compact commutative group G. Every function f ϵ P k c ( G ) is definitizable in the sense thatis positive definite for certain complex measures ω on G with finite support [9]. The proof of this fact was base on a result of M. A. Naimark about common nonpositive eigenvectors of commuting unitary operators in a Pontrjagin space. It is the aim of this note to prove without any use of the theory of Pontrjagin spaces the definitizability of functions f ϵ P k c ( G ) which are of polynomial growth. In Section 3 we show, how the definitizability of functions f ϵ P k c ( G ) can be used to prove the existence of common non‐positive eigenvectors of commuting unitary operators in a Pontrjagin space.