Numerical range of values

The numerical value range (English: numerical range ) is a term from the mathematical branch of functional analysis . A set of the basic field is assigned to a continuous linear operator or more generally to an element of a Banach algebra . This set combines algebraic information with properties of the norm .

Definitions

This article uses the basic field of complex numbers; In the case of the real numbers, complications arise in some places, which are hidden here for the sake of simplicity. Let it be a normalized algebra over with unity . A continuous, linear functional is called a state on , if , and let it be the set of all states on ; according to Hahn-Banach's theorem, this is not empty. For an element is called ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle e}$ ${\displaystyle f\colon A\rightarrow \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle \|f\|=1=f(e)}$ ${\displaystyle D(A,e)}$ ${\displaystyle A}$ ${\displaystyle a\in A}$

${\displaystyle V(A,a):=\{f(a);\,f\in D(A,e)\}\,\subset \,\mathbb {C} }$

the numerical range of . ${\displaystyle a}$

Since a convex and, according to the Banach-Alaoglu theorem, weak- * - compact subset of the dual space , the numerical range of values ​​must also be a convex and compact subset in . thats why ${\displaystyle D(A,e)}$ ${\displaystyle A'}$ ${\displaystyle \mathbb {C} }$

${\displaystyle v(A,a):=\sup\{|\lambda |;\,\lambda \in V(A,a)\}}$

a finite number, called the numerical radius of . ${\displaystyle a}$

A common numerical range of values ​​is defined for several elements using the formula ${\displaystyle a_{1},\ldots ,a_{n}\in A}$

${\displaystyle V(A,a_{1},\ldots ,a_{n}):=\{(f(a_{1}),\ldots ,f(a_{n}));\,f\in D(A,e)\}\subset \mathbb {C} ^{n}}$

and this is also a convex and compact set.

Sub-algebras

One can show that the numerical range of values ​​does not depend on the surrounding algebra, that is, one can move on to smaller or larger algebras as long as they only contain the element and the elements or . This is mainly due to the fact that states on subalgebras can be continued on the larger algebra because of the Hahn-Banach theorem on states. In particular, with a normalized algebra you can go over to completion without changing the numerical range of values. ${\displaystyle a}$ ${\displaystyle a_{1},\ldots ,a_{n}\in A}$

Numerical index

It is easy to show that the numerical radius is a semi-norm ; but it is even a norm because it applies

If a complex normalized algebra then applies: ${\displaystyle A}$

${\displaystyle {\frac {1}{e}}\|a\|\leq v(A,a)\leq \|a\|}$for everyone . ${\displaystyle a\in A}$

It is the Euler number . thats why ${\displaystyle e}$

${\displaystyle n(A):=\inf\{v(A,a);\,a\in A,\|a\|=1\}}$

is a number from the interval and is called the numeric index of . For commutative C * -algebras the numerical index is always , for any C * -algebras one can show that the numerical index is greater than or equal to . ${\displaystyle [{\tfrac {1}{e}},1]}$ ${\displaystyle A}$ ${\displaystyle 1}$ ${\displaystyle {\tfrac {1}{2}}}$

Comparison with the spectrum

The numerical range of values ​​depends not only on the algebraic structure of the algebra under consideration , but also on the norm via the state space. If one goes over to an equivalent norm with , forms the state space with respect to and from this the numerical value range, one possibly obtains a different set, which is therefore more precisely designated with or . Let the set of all equivalent algebra norms with . ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle p(e)=1}$ ${\displaystyle p}$ ${\displaystyle V_{p}(A,a)}$ ${\displaystyle V_{p}(A,a_{1},\ldots ,a_{n})}$ ${\displaystyle {\mathcal {N}}_{A}}$ ${\displaystyle p(e)=1}$

The spectrum of an element or the common spectrum of finitely many commuting elements of a complex Banach algebra, on the other hand, depends only on the algebraic structure and is retained when changing to an equivalent norm. It is therefore astonishing that the following relationship exists, where the convex hull denotes: ${\displaystyle \sigma _{A}(a)}$ ${\displaystyle \sigma _{A}(a_{1},\ldots ,a_{n})}$ ${\displaystyle \mathrm {conv} }$

Let it be a complex Banach algebra, and if commuting elements are off , then the following applies: ${\displaystyle A}$ ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle A}$

${\displaystyle \mathrm {conv} \,\sigma _{A}(a_{1},\ldots ,a_{n})=\bigcap _{p\in {\mathcal {N}}_{A}}V_{p}(A,a_{1},\ldots ,a_{n})}$.

For the spectrum and the numerical range of values ​​of an element , the following formulas also apply for the maximum of the real parts : ${\displaystyle a\in A}$

${\displaystyle {\begin{aligned}\max\{\operatorname {Re} \lambda ;\,\lambda \in \sigma _{A}(a)\}\;\;\;&=\inf \left\{{\frac {1}{\alpha }}\ln \|\exp(\alpha a)\|;\,\alpha >0\right\}\\&=\lim _{\alpha \to \infty }{\frac {1}{\alpha }}\ln \|\exp(\alpha a)\|\\\max\{\operatorname {Re} \lambda ;\,\lambda \in V(A,a)\}&=\sup \left\{{\frac {1}{\alpha }}\ln \|\exp(\alpha a)\|;\,\alpha >0\right\}\\&=\lim _{\alpha \to 0}{\frac {1}{\alpha }}\ln \|\exp(\alpha a)\|\\&=\inf \left\{{\frac {1}{\alpha }}(\|1+\alpha a\|-1);\,\alpha >0\right\}\\&=\lim _{\alpha \searrow 0}{\frac {1}{\alpha }}(\|1+\alpha a\|-1)\end{aligned}}}$

It should be noted about these formulas that in every Banach algebra the exponential series converges to a different element and therefore the natural logarithm can be formed. ${\displaystyle \exp(a):=\sum _{k=0}^{\infty }{\frac {1}{k!}}a^{k}}$ ${\displaystyle 0}$ ${\displaystyle \ln \|\exp(a)\|}$

Hermitian elements

If a C * -algebra, then self-adjoint elements , i.e. those that satisfy, have a real spectrum, but the converse does not apply. This is different when you move from the spectrum to the numerical value range. Therefore, it seems reasonable to see a generalization of self-adjoint elements in the elements of any complex Banach algebra with one element, the numerical range of which lies in the real numbers. Such elements are called Hermitian ; they play an important role in Vidav-Palmer's theorem , which characterizes the C * algebras among the Banach algebras. ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle a^{*}=a}$

Versions for operators

The concept of the numerical value range goes back to forerunners for operators on normalized spaces. Let be a normalized space and an element from the Banach algebra of bounded linear operators on . Then one can form the numerical range of values ​​defined above for the element of the Banach algebra . For Hilbert spaces has Otto Toeplitz in 1918 the amount ${\displaystyle X}$ ${\displaystyle T\in L(X)}$ ${\displaystyle X}$ ${\displaystyle V(L(X),T)}$ ${\displaystyle T}$ ${\displaystyle L(X)}$ ${\displaystyle X}$

${\displaystyle W(T):=\{\langle Tx,x\rangle ;\,x\in X,\|x\|=1\}}$

considered, see also the article Numerical Range of Values ​​(Hilbert Space) . This can be generalized to any normalized spaces by replacing the scalar product with a semi-inner product and ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle [\cdot ,\cdot ]}$

${\displaystyle W(T):=\{[Tx,x];\,x\in X,[x,x]=1\}}$

Are defined. Friedrich L. Bauer examined the crowd in 1962

${\displaystyle V(T):=\{f(Tx);\,x\in X,f\in X',f(x)=1,\|x\|=1,\|f\|=1\}}$

initially only in finite-dimensional spaces, but the same definition can also be used for any normalized spaces. There is the following relationship between these terms:

Let be a normalized space and , then: ${\displaystyle X}$ ${\displaystyle T\in L(X)}$

${\displaystyle W(T)\subset V(T)\subset V(L(X),T)={\overline {\mathrm {conv} \,W(T)}}}$.

For normalized spaces one can use the numerical index

${\displaystyle n(X):=\inf\{v(L(X),T);\,T\in L(X),\|T\|=1\}}$

define, which is nothing more than the numerical index of the Banach algebra and therefore also a number from the interval . For Hilbert spaces of the dimension greater than or equal to , one can show that their numerical index is the same . The Banach space of continuous functions on the compact Hausdorff space has the numerical index . ${\displaystyle L(X)}$ ${\displaystyle [{\tfrac {1}{e}},1]}$ ${\displaystyle 2}$ ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle C(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle 1}$

literature

• Frank Bonsall & John Duncan: Complete Normed Algebras (= results of mathematics and its border areas / NF Vol. 80). Springer, Berlin 1973, ISBN 3-540-06386-2 .
• Frank Bonsall & John Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras (= London Mathematical Society: Lecture Note Series. Vol. 2). CUP, London 1971, ISBN 0-521-07988-8 .
• Otto Toeplitz: The algebraic analogue to a sentence by Fejer . In: Mathematical Journal . Vol. 2, 1918
• FL Bauer : On the field of values ​​subordinate to a norm. In: Numerische Mathematik. Vol. 4, 1962, DOI: 10.1007 / BF01386300 , pp. 103-113

Footnotes

1. ^ FF Bonsall, J. Duncan: Complete Normed Algebras , §10, Definition 1.
2. ↑ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Chapter 1, §2, Definition 1.
3. ↑ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Chapter 1, §2, Definition 11.
4. ↑ FF Bonsall, J. Duncan: Complete Normed Algebras , §10, Theorem 14.
5. ↑ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , end of Chapter 1, §4.
6. ↑ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Chapter 1, §2, Theorem 13.
7. ↑ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras , Chapter 1, §2 and §3.
8. ↑ Otto Toeplitz: The algebraic analogue to a sentence by Fejer , pages 187-197.
9. ↑ FL Bauer: On the field of values ​​subordinate to a norm. In: Numerische Mathematik. Vol. 4, 1962, DOI: 10.1007 / BF01386300 , pp. 103-113
10. ^ FF Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras. Chapter 1, §9 Theorems 4 and 8.