# Covariance operator

__: Operator in probability theory__

**Short description**In probability theory, for a probability measure **P** on a Hilbert space *H* with inner product [math]\displaystyle{ \langle \cdot,\cdot\rangle }[/math], the **covariance** of **P** is the bilinear form Cov: *H* × *H* → **R** given by

- [math]\displaystyle{ \mathrm{Cov}(x, y) = \int_{H} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \mathbf{P} (z) }[/math]

for all *x* and *y* in *H*. The **covariance operator** *C* is then defined by

- [math]\displaystyle{ \mathrm{Cov}(x, y) = \langle Cx, y \rangle }[/math]

(from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is
self-adjoint (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). When **P** is a centred Gaussian measure, *C* is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace.

Even more generally, for a probability measure **P** on a Banach space *B*, the covariance of **P** is the bilinear form on the algebraic dual *B*^{#}, defined by

- [math]\displaystyle{ \mathrm{Cov}(x, y) = \int_{B} \langle x, z \rangle \langle y, z \rangle \, \mathrm{d} \mathbf{P} (z) }[/math]

where [math]\displaystyle{ \langle x, z \rangle }[/math] is now the value of the linear functional *x* on the element *z*.

Quite similarly, the covariance function of a function-valued random element (in special cases is called random process or random field) *z* is

- [math]\displaystyle{ \mathrm{Cov}(x, y) = \int z(x) z(y) \, \mathrm{d} \mathbf{P} (z) = E(z(x) z(y)) }[/math]

where *z*(*x*) is now the value of the function *z* at the point *x*, i.e., the value of the linear functional [math]\displaystyle{ u \mapsto u(x) }[/math] evaluated at *z*.

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Original source: https://en.wikipedia.org/wiki/Covariance operator.
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