Positive operator

positive mapping

A positive operator on a Hilbert space is a linear operator $ A $ for which the corresponding quadratic form $ ( Ax, x) $ is non-negative. A positive operator on a complex Hilbert space is necessarily symmetric and has a self-adjoint extension that is also a positive operator. A self-adjoint operator $ A $ is positive if and only if any of the following conditions holds: a) $ A = B ^ {*} B $, where $ B $ is a closed operator; b) $ A = B ^ {2} $, where $ B $ is a self-adjoint operator; or c) the spectrum of $ A $( cf. Spectrum of an operator) is contained in $ [ 0, \infty ) $. The set of positive bounded operators on a Hilbert space forms a cone in the algebra of all bounded operators.

A positive operator on a vector space $ X $ containing a cone $ K $ is a mapping from $ X $ into itself that preserves the given cone $ K $ in $ X $. Integral operators with positive kernels on various function spaces with given cones of positive functions are positive linear operators. Subject to certain additional conditions on the geometry of the cone $ K $ and the action of the positive operator $ A $, one can establish the existence of eigen vectors of $ A $ in $ X $( the corresponding eigen values are called positive or leading ones, as they exceed the absolute values of all the other eigen values). For example, it has been shown

that if $ A $ is a positive completely-continuous operator with a non-zero spectrum, then its spectral radius is a positive eigen value. The condition of compactness may be replaced by conditions on the behaviour of the resolvent .

In the case of positive non-linear operators one examines the existence of a fixed point (i.e. a solution to the equation $ Ax = x $) and the possibility of finding this point as the limit of certain recurrent sequences.

Some results from the theory of positive operators can be transferred to operators that leave invariant given subsets of more general type than a cone .

A positive operator on an involution algebra (a $ * $- algebra) $ A $ is a linear mapping from $ A $ into an involution algebra $ B $ which transfers positive elements to positive elements. The most studied are the positive operators on a $ C ^ {*} $- algebra (these are a particular case of positive operators on a space with a cone because the positive elements in a $ C ^ {*} $- algebra form a cone). Schwartz's inequality holds for positive operators on $ C ^ {*} $- algebras: $ \phi ( a ^ {2} ) \geq ( \phi ( a)) ^ {2} $ if $ a = a ^ {*} $. The extreme points have been found for the set of unitary positive operators (i.e. the ones that preserve the unit element). Studies have also been made on positive completely-continuous operators, i.e. linear mappings $ \phi : A \rightarrow B $ for which all the mappings

$$ ( a _ {ij} ) _ {i,j= 1 } ^ {n} \rightarrow ( \phi ( a _ {ij} )) _ {i,j= 1 } ^ {n} $$

of the matrix $ C ^ {*} $- algebra $ M( A) $ into $ M( B) $ are positive. An analogue of the theorem on the extension of a positive functional applies for positive completely-continuous operators: A positive completely-continuous operator on a $ C ^ {*} $- algebra $ A $ into a certain von Neumann algebra can be extended to a positive completely-continuous operator on any $ C ^ {*} $- algebra containing $ A $. If one of the $ C ^ {*} $- algebras $ A $ and $ B $ is commutative (and only in that case), then any positive operator is completely continuous.

A positive operator on a Banach space $ E $ is a linear operator $ A $ such that $ AK \subset K $, where $ K $ is a positive cone in $ E $. An eigen vector of $ A $ lying in $ K $ is called positive, and the corresponding eigen value is positive. If $ K $ is a reproducing cone while $ A $ is a positive completely-continuous operator and $ A ^ {p} u \geq \alpha u $ for a certain vector $ u $ not belonging to $ K $, with $ p $ a natural number and $ \alpha > 0 $, then the spectral radius $ r _ {A} $ of $ A $ is a positive eigen value of $ A $; moreover, $ r _ {A} \geq \alpha ^ {1/p} $( the Krein–Rutman theorem).

References

[1]	N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1981) (Translated from Russian)
[2]	S. Sherman, "Order in operator algebras" Amer. J. Math. , 73 : 1 (1951) pp. 227–232
[3]	M.G. Krein, M.A. Rutman, "Linear operators leaving invariant a cone in a Banach space" Transl. Amer. Math. Soc. (1) , 10 (1962) pp. 199–325 Uspekhi Mat. Nauk , 3 : 1 (1948) pp. 3–95
[4]	H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
[5]	M.A. Krasnosel'skii, A.V. Sobolev, "On cones of finite rank" Soviet Math. Dokl. , 16 : 6 (1975) pp. 1621–1625 Dokl. Akad. Nauk SSSR , 225 : 6 (1975) pp. 1256–1259
[6]	M.A. Krasnosel'skii, et al., "Integral operators and spaces of summable functions" , Noordhoff (1967) (Translated from Russian)
[7]	J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
[8]	M.A. Krasnosel'skii, "Positive solutions of operator equations" , Wolters-Noordhoff (1964) (Translated from Russian)

References

[a1]	N. Dunford, J.T. Schwartz, "Linear operators. Spectral theory" , 2 , Wiley (Interscience) (1988) pp. 906ff
[a2]	M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972) pp. 195ff
[a3]	B.Z. Vulikh, "Functional analysis for scientists and technologists" , Pergamon (1963) pp. Sect. 13.6 (Translated from Russian)