Pencils of pairs of projections
Miaomiao Cui, Guoxing Ji

TL;DR
This paper characterizes when a self-adjoint operator can be expressed as a linear combination of two projections at specific real parameters, providing conditions, representations, and uniqueness results for such decompositions.
Contribution
It offers a complete characterization of operators as pencils of projections at certain parameters, including representations, connectivity of pairs, and uniqueness conditions.
Findings
Characterizes when T = λP + Q for projections P, Q.
Provides a representation for all such pairs (P, Q).
Determines the uniqueness of the real parameter λ.
Abstract
Let be a self-adjoint operator on a complex Hilbert space . We give a sufficient and necessary condition for to be the pencil of a pair of projections at some point . Then we represent all pairs of projections such that for a fixed , and find that all such pairs are connected if . Afterwards, the von Neumann algebra generated by such pairs is characterized. Moreover, we prove that there are at most two real numbers such that is the pencils at these real numbers for some pairs of projections. Finally, we determine when the real number is unique.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Spectral Theory in Mathematical Physics
