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The operator is said to be positive-definite, and written >, if , >, for all {}. [ 1 ] Many authors define a positive operator A {\displaystyle A} to be a self-adjoint (or at least symmetric) non-negative operator.
In other words, a positive linear operator maps the positive cone of the domain into the positive cone of the codomain. Every positive linear functional is a type of positive linear operator. The significance of positive linear operators lies in results such as Riesz–Markov–Kakutani representation theorem.
Positive maps are monotone, i.e. () for all self-adjoint elements ,. Since ‖ ‖ ‖ ‖ for all self-adjoint elements , every positive map is automatically continuous with respect to the C*-norms and its operator norm equals ‖ ‖.
In functional analysis and quantum information science, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalization of quantum ...
The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts — see the Hahn decomposition theorem .
The off-diagonal elements are complex conjugates of each other (also called coherences); they are restricted in magnitude by the requirement that () be a positive semi-definite operator, see below. A density operator is a positive semi-definite, self-adjoint operator of trace one acting on the Hilbert space of the system.
Operator. Greetings, and welcome to the Plug Power fourth-quarter 2024 earnings call. [Operator instructions] As a reminder, this conference is being recorded. It is now my pleasure to introduce ...
Conversely, every operator-valued positive-definite function arises in this way. Recall that every (continuous) scalar-valued positive-definite function on a topological group induces an inner product and group representation φ(g) = 〈U g v, v〉 where U g is a (strongly continuous) unitary representation (see Bochner's theorem).