Where does operator collide?

In mathematics an operator is a mapping that sends the elements of one space to another, most often a function to another function. In Methods of Modern Mathematical Physics, Reed and Simon set out the theory of linear operators on Hilbert and Banach spaces, where an operator is defined by how it acts on the elements of its domain and studied through its norm, its spectrum, and its algebra. The referent is a rule of correspondence between spaces, not a number and not a thing in the world. A differentiation operator or an integral operator is an object you compose, invert, and take the spectrum of. This meaning shares only its name with the operator of genetics, and it differs from the quantum observable, which lays a physical interpretation on top of the same mathematics.[1]

Unit

A mapping from one space of functions or vectors to another

Recovery means

Not applicable

Measured by

Defined by its action on the elements of its domain, and studied through its norm, spectrum, and algebraic properties

In quantum mechanics an observable, such as position, momentum, or energy, is represented by a Hermitian operator acting on the state vectors of a Hilbert space. Dirac's The Principles of Quantum Mechanics gives the standard account: each measurable quantity corresponds to a linear operator, the possible results of measuring it are that operator's eigenvalues, and the state fixes the probability of each. The referent is a physical quantity you can measure, tied to an operator by a rule of representation. This meaning borrows the mathematics of the formal operator but is not the same construct: the mathematics is a tool, while the observable adds the claim that eigenvalues are outcomes and that Hermiticity keeps them real. It has nothing to do with the operator of the operon or with a telephone operator.[2]

Unit

A physical observable represented on the state space

Recovery means

Not applicable

Measured by

Represented by a Hermitian operator whose eigenvalues are the possible outcomes of measuring the quantity

In molecular genetics the operator is a short segment of DNA next to a gene where a repressor protein binds to switch transcription off. Jacob and Monod introduced it in Genetic Regulatory Mechanisms in the Synthesis of Proteins, their model of the lac operon, where the operator sits between the promoter and the structural genes and control turns on whether the repressor is bound to it. The referent is a physical stretch of nucleotides with a location on the chromosome, not a mapping and not a person. Here the word names a place where regulation happens, defined by what binds there, which makes it a different construct from the mathematical mapping and from the network operator of telecommunications.[3]

Unit

A segment of regulatory DNA

Recovery means

Not applicable

Measured by

Located on the DNA by binding and deletion studies as the site where repressor binding blocks transcription