Positive operator

Positive operator is a term from functional analysis that is used in two different ways. On the one hand, a Hilbert space operator or an element of a C ^* algebra can be positive in terms of spectral theory . On the other hand, operators between ordered vector spaces are called positive if they contain the order structure. Both terms have great meanings in mathematics, as illustrated in examples.

Positive Hilbert space operators

Let a - Hilbert space with a scalar product . The following statements are equivalent for a linear continuous operator : ${\displaystyle H}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle P:H\rightarrow H}$

• For all true . ${\displaystyle x\in H}$ ${\displaystyle \langle Px,x\rangle \geq 0}$
• ${\displaystyle P}$is self adjoint and for everyone . ${\displaystyle \langle Px,x\rangle \geq 0}$ ${\displaystyle x\in H}$
• ${\displaystyle P}$is self adjoint and the spectrum of lies in . ${\displaystyle P}$ ${\displaystyle {\mathbb {R} }_{0}^{+}}$
• There is a continuous linear operator with . ${\displaystyle A\colon H\rightarrow H}$ ${\displaystyle P=A^{*}A}$
• There is a self-adjoint operator with . ${\displaystyle A\colon H\rightarrow H}$ ${\displaystyle P=A^{2}}$

An operator that has one and therefore all of these properties is called positive . The equivalence of point 1 and 2 follows from the polarization formula for sesquilinear forms and only works for complex Hilbert spaces and not for real ones. If the Hilbert space is finite-dimensional, the operators can be represented as matrices . The definition of positivity given here corresponds to the positivity known from linear algebra , i.e. a matrix is ​​positive if it can be diagonalized and all eigenvalues ​​are not negative. Positive matrices play an important role in determining extreme values in the multidimensional case.

In the above list of equivalent characterizations, only the first statement makes direct reference to Hilbert space elements. The other three statements can be carried over directly to C ^* algebras . The relationship to the first characterization is preserved, since every C ^* -algebra can be understood as a subalgebra of the C ^* -algebra of the operators on a Hilbert space according to the Gelfand-Neumark theorem .

In the commutative C ^* -algebra of the continuous functions on a locally compact space that vanish at infinity, the positive elements are exactly those functions whose image lies in. ${\displaystyle C_{0}(X)}$ ${\displaystyle {\mathbb {R} }_{0}^{+}}$

The positive elements of a C ^* -algebra form a cone and therefore represent an essential structural element. They play an important role in the decomposition of the polar . The C * algebra receives an order structure through the definition: is positive. This leads on to the next term, positive operators. ${\displaystyle A\leq B:\Leftrightarrow B-A}$

Positive operators between order structures

Vector spaces E with a partial order are called an ordered vector space . Most of the time, one still demands that this order structure is compatible with the linear vector space structure, i.e. This means that for with and always and applies. ${\displaystyle x,y\in E}$ ${\displaystyle x,y\geq 0}$ ${\displaystyle \lambda \in {\mathbb {R} }_{0}^{+}}$ ${\displaystyle x+y\geq 0}$ ${\displaystyle \lambda x\geq 0}$

Examples of such ordered vector spaces are:

• ${\displaystyle \mathbb {R} }$ with the usual order structure.
• ${\displaystyle {\mathbb {R} }^{n}}$, where if and only if for all . ${\displaystyle (x_{1},\ldots ,x_{n})\leq (y_{1},\ldots ,y_{n})}$ ${\displaystyle x_{i}\leq y_{i}}$ ${\displaystyle 1\leq i\leq n}$
• L ^p ([0,1]) , where , if for almost all . ${\displaystyle f\leq g}$ ${\displaystyle f(x)\leq g(x)}$ ${\displaystyle x\in [0,1]}$
• A C ^* algebra with the order structure defined above.

An operator between ordered vector spaces is called positive or monotonic if it always follows, i.e. H. when the order structures get. ${\displaystyle P:E\rightarrow F}$ ${\displaystyle x\geq 0}$ ${\displaystyle Px\geq 0}$ ${\displaystyle P}$

A well-known example is the Bernstein operator on , which assigns its th Bernstein polynomial to every continuous function . Is (point by point), then is also (point by point), as you can easily see from the formula for . Such positive operators play an important role in approximation theory , for example in Korowkin's theorem . ${\displaystyle P_{n}}$ ${\displaystyle C[0,1]}$ ${\displaystyle n}$ ${\displaystyle f\geq 0}$ ${\displaystyle P_{n}(f)\geq 0}$ ${\displaystyle \textstyle P_{n}(f)(t)=\sum _{i=0}^{n}{\tbinom {n}{i}}\cdot f\left({\tfrac {i}{n}}\right)\cdot t^{i}\,(1-t)^{n-i}}$ ${\displaystyle t\in [0,1]}$

Both positivity terms appear in the following example. According to the above, a C ^* -algebra is an ordered space. The set of complex numbers is also a C ^* algebra, which is the cone of positive elements . A continuous linear functional is called positive if it is a positive operator between the ordered spaces. So it is positive if for everyone . These positive functionals play a central role in Gelfand-Neumark's theorem . ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\mathbb {R} }_{0}^{+}}$ ${\displaystyle f:{\mathcal {A}}\rightarrow {\mathbb {C} }}$ ${\displaystyle f}$ ${\displaystyle f(A^{*}A)\geq 0}$ ${\displaystyle A\in {\mathcal {A}}}$

See also

• Monotonic operator
• Completely positive operator

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• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras (1983)
• GJO Jameson: Ordered Linear Spaces , Springer Lecture Notes no.141 (1970)