Numerical range

The numerical value range (English: numerical range) is a term from the mathematical branch of functional analysis. A continuous linear operator, or more generally an element of a Banach algebra is associated with a lot of the body. This set combines information with algebraic properties of the standard.

Definitions

In this article, the basic field of complex numbers is used; in the case of real numbers arise in some places complications that are hidden here for simplicity. It is a normed algebra with identity over. A continuous linear functional is called a state if, and it is the set of all states on; is not empty, by the theorem of Hahn- Banach. For an element called

The numerical range of.

Since a convex and by the theorem of Banach - Alaoglu weak - * - compact subset of the dual space is also the numerical value range must be a convex and compact subset in his. is therefore

A finite number, it is called numerical radius.

For several elements to define a common numeric range by the formula

And this is also a convex and compact set.

Subalgebras

It can be shown that the numeric value range is not dependent on the surrounding algebra, that is, one can proceed to smaller or larger algebras, as long as such, or only the one member and the elements. This is mainly due to that it is possible for states to subalgebras of the set of Hahn-Banach continue to states on the larger algebra. In particular, you can pass in a normed ring to complete, without changing the numerical value range thereby.

Numeric Index

Easy, one can show that the numerical radius is a semi-norm; but it is as much as a standard, because it is

This is the Euler number. is therefore

A number from the interval and is called the numeric index of. For commutative C *-algebras the numeric index is always, for any C *-algebras, one can show that the numeric index is greater than or equal.

Compared to the spectrum

The numerical range of values ​​depends not only on the structure of the considered algebraic algebra, but also through the state space of the standard. Going to an equivalent norm with over forms the state space with respect to and from the numerical range of values, so you may receive a different amount, which is therefore more accurately referred to as or. Next is with the set of all equivalent Algebrennormen.

The spectrum of an element or the common range of a finite number of commuting elements of a complex Banach algebra, however, depends only on the algebraic structure and is maintained during the transition to an equivalent standard. Therefore, it is surprising that the following relationship exists, the convex hull of call:

Moreover, the following formulas for the maximum of the real parts apply to the spectrum and the numerical range of an element:

Note these formulas that in every Banach algebra, the exponential series converges to an element that is different from, and therefore the natural logarithm can be formed.

Hermitian elements

Is a C *-algebra, so have self-adjoint elements, ie, those which satisfy known to be a real spectrum, however, shall not apply the inversion. This is different when one passes from the range to the numerical range of values. Therefore, it is natural in the elements of an arbitrary complex Banach algebra with unit element, whose numerical value range is the real numbers to see a generalization of self-adjoint elements. They are called hermitian elements, they play an important role in the set of Vidav - Palmer, which characterizes the C *-algebras among Banach algebras.

Versions for operators

The concept of numerical range of values ​​goes back to precursors of operators on normed spaces. Be a normed space and an element of the Banach algebra of bounded linear operators on. Then one can form the above-defined numeric range of the element of the Banach algebra. For Hilbert spaces Otto Toeplitz did in 1918, the amount

Considered, see also the article Numeric value range ( Hilbert space ). This can be generalized to arbitrary normed spaces by the dot product is replaced by a semi- inner product, and

Defined. Friedrich L. Bauer examined in 1962 the amount

Initially only in finite dimensional spaces, but the same definition can be used for arbitrary normed spaces. The following relationship exists between these terms:

For normed spaces can be the numeric index

Define the order is nothing more than the numerical index of Banach algebra and therefore also a number from the interval. For Hilbert spaces of dimension greater than or equal, one can show that their numerical index is equal. The Banach space of continuous functions on the compact Hausdorff space has numerical index.