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International Journal of Theoretical Physics

Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Marcel Polakovič, Zdenka Riečanová

Published Aug 11, 2010• 24 citations
Publisher
DOI
10.1007/s10773-010-0458-3

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The Bender-Dunne basis operators, T−m,n=2−n∑k=0n(nk)qkp−mqn−k\mathsf {T}_{-m,n}=2^{-n}\sum _{k=0}^n { n \choose k} \mathsf {q}^k \mathsf {p}^{-m} \mathsf {q}^{n-k}T−m,n​=2−n∑k=0n​(kn​)qkp−mqn−kT−m,n=2−n∑k=0nnkqkp−mqn−k where q\mathsf {q}qq and p\mathsf {p}pp are the position and momentum operators, respectively, are formal integral operators in position representation in the entire real line R\mathbb {R}RR for positive integers n and m. We show, by explicit construction of a dense domain, that the operators T−m,n\mathsf {T}_{-m,n}T−m,n​T−m,n's are densely defined operators in the Hilbert space L2(R)L^2(\mathbb {R})L2(R)L2(R).