operator0

Operator0 is the zero operator on a vector space V over a field F. It is the linear map O0: V → V defined by O0(v) = 0 for every v in V. In standard notation, it is often denoted 0_V or simply 0 when the context is clear.

Properties: O0 is linear; its kernel is V and its range is {0}. Its norm is 0 in normed spaces, and it is idempotent since O0 ∘ O0 = O0. It is not invertible unless V is the zero vector space. In finite dimensions it corresponds to the zero matrix.

Examples: On R^n, O0(x) = 0 for all x. On the space C[0,1] of continuous functions, (O0 f)(t) = 0 for all t in [0,1].

Role and applications: The zero operator provides a neutral baseline in operator theory and linear algebra, useful in proving existence and structure results or in decompositions of operators, such as writing an operator A as A = A1 + O0 when separating a nontrivial part from the trivial part. Spectrally, the spectrum of O0 is {0}, and the only eigenvalue is 0 with multiplicity equal to the dimension of V.

See also: identity operator, projection operator, nilpotent operator, bounded linear operator. References: standard texts in linear algebra and functional analysis introduce the zero operator as a basic, well-understood example of a linear map.