Numerical range of values ​​(Hilbert space)

The numerical range of values is a term from functional analysis and linear algebra .

definition

For a complex Hilbert space with scalar product and a bounded linear operator , the numerical range of is given by ${\displaystyle H}$ ${\displaystyle \langle {\cdot },{\cdot }\rangle }$ ${\displaystyle T\colon H\to H}$ ${\displaystyle T}$

${\displaystyle W(T):=\left\{\langle Tx,x\rangle :\lVert x\rVert =1\right\},}$

where is the norm induced by on . ${\displaystyle \lVert {\cdot }\rVert }$ ${\displaystyle \langle {\cdot },{\cdot }\rangle }$ ${\displaystyle H}$

Analogous to the spectral radius , the numerical radius is defined by . ${\displaystyle w(T):=\sup\{|\lambda |:\lambda \in W(T)\}}$

In the special case of complex-valued, square matrices , the definition of the numerical value range is equivalent to ${\displaystyle A\in \mathbb {C} ^{n\times n}}$

${\displaystyle W(A)=\left\{{\frac {x^{*}Ax}{x^{*}x}}:x\in \mathbb {C} ^{n}\setminus \{0\}\right\}.}$

${\displaystyle W(A)}$is here the image area of the Rayleigh quotient .

properties

The following properties apply to constrained linear operators . ${\displaystyle T\colon H\to H}$

• ${\displaystyle W(T)\subseteq \{\lambda \in \mathbb {C} :|\lambda |\leq \lVert T\rVert \}}$or equivalent to it . Here denotes the operator norm of . ${\displaystyle w(T)\leq \lVert T\rVert }$ ${\displaystyle \lVert T\rVert }$ ${\displaystyle T}$
• The numerical range of is convex . (Toeplitz-Hausdorff theorem) ${\displaystyle T}$
• The spectrum is in the final of : . Is finite-dimensional, even applies . ${\displaystyle \sigma (T)}$ ${\displaystyle W(T)}$ ${\displaystyle \sigma (T)\subseteq {\overline {W(T)}}}$ ${\displaystyle H}$ ${\displaystyle \sigma (T)\subseteq W(T)}$
• Anything for which is true is an eigenvalue of . ${\displaystyle \lambda \in W(T)}$ ${\displaystyle |\lambda |=\lVert T\rVert }$ ${\displaystyle T}$

Applications

The right real axis intercept of the numerical value range is the logarithmic norm , for a matrix this is ${\displaystyle A}$

${\displaystyle \mu (A)=\max \left\{x^{*}Ax:\ x^{*}x=1\right\}.}$

With it a limit for the spectral norm of the matrix exponential can be given, it applies

${\displaystyle \|e^{tA}\|_{2}\leq e^{t\mu (A)},\ t\geq 0.}$

Because solves the initial value problem . Then it holds for the Euclidean norm that its derivative satisfies the inequality , from which it follows. This corresponds to the limit for the matrix exponential. ${\displaystyle y(t)=e^{tA}y_{0}}$ ${\displaystyle y'(t)=Ay(t),\,y(0)=y_{0}}$ ${\displaystyle N(t)=\|y(t)\|^{2}}$ ${\displaystyle N'(t)=2y(t)^{*}y'(t)=2y(t)^{*}Ay(t)\leq 2\mu (A)y(t)^{*}y(t)=\mu (A)N(t)}$ ${\displaystyle N(t)\leq e^{2t\mu (A)}N(0)}$

literature

• E. Hairer, G. Wanner, Solving ordinary differential equations II , Springer, 1991.