Resolvent

This article explains the resolvent in functional analysis , for the resolvent in logic see Resolution (logic) and for resolvent in algebra see Lagrange resolvent .

In mathematics and theoretical physics , the resolvent (sometimes called Greenscher's operator ) is the inverse of a linear operator or a matrix that is shifted with a complex number . The set of values for which this inverse is well defined is the operator's resolvent set ; the complement of this set is its spectrum . Applications concern all aspects of operator theory in functional analysis , especially perturbation calculation . ${\displaystyle z}$ ${\displaystyle z}$

definition

For a linear operator (or a matrix ), the resolvent set is defined as the complement of the spectrum of , i. H. as the set of all complex numbers for which the operator is restrictedly invertible. The resolvent set is open as the complement of the spectrum . The resolvent is defined on the resolvent set ${\displaystyle A}$ ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ ${\displaystyle \rho (A)}$ ${\displaystyle A}$ ${\displaystyle z}$ ${\displaystyle zI-A}$

${\displaystyle R\left(A,z\right)=(zI-A)^{-1}\ .}$

Many authors use the definition of the resolvent , which only inverts the sign . ${\displaystyle R\left(A,z\right)=(A-zI)^{-1}}$

Properties and uses

The resolvent is an operator-valued analytical function and can be represented by Neumann's series , where the spectral radius is : ${\displaystyle \{z\in \mathbb {C} :|z|>r\}}$ ${\displaystyle r}$

${\displaystyle R\left(A,z\right)=\sum _{n=0}^{\infty }{\frac {A^{n}}{z^{n+1}}}}$.

The resolvent will u. a. used to describe eigenvalue expansions of perturbed operators, for example the expansions of Rellich - Kato and Strutt - Schrödinger .

Resolvent Identities

The first and second resolvent identities are useful for calculations. The first resolvent identity follows from inversion ${\displaystyle \left(z_{1}-z_{2}\right)I=z_{1}I-A-(z_{2}I-A)}$

${\displaystyle R\left(A,z_{2}\right)-R(A,z_{1})=(z_{1}-z_{2})R(A,z_{1})R(A,z_{2})=(z_{1}-z_{2})R(A,z_{2})R(A,z_{1}),}$

and the second resolvent identity follows by means of inversion ${\displaystyle A_{1}-A_{2}=zI-A_{2}-\left(zI-A_{1}\right)}$

${\displaystyle R\left(A_{1},z\right)-R(A_{2},z)=R(A_{1},z)(A_{1}-A_{2})R(A_{2},z).}$

literature

• Dirk Werner : functional analysis , Springer-Verlag, Berlin, 2007, ISBN 978-3-540-72533-6 .