# Resolvent

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## resolvent

[ri′zäl·vənt]*T*on a Banach space, the function, defined on the complement of the spectrum of

*T*given by (

*T*- λ

*I*)

^{-1}for each λ in this complement, where

*I*is the identity operator; this enables a study of

*T*relative to its eigenvalues.

## Resolvent

a mathematical term with various meanings. We speak, for example, of resolvent equations, resolvent kernels, and resolvent operators.

In algebra, the term “resolvent” is used in several senses. Thus, by the resolvent of the algebraic equation *f*(*x*) = 0 of degree *n* we mean an algebraic equation *g*(*x*) = 0 such that its coefficients are rational functions of the coefficients of *f*(*x*) and, if the roots of this equation are known, it is possible to find the roots of *f*(*x*) = 0 by solving simpler equations of degree at most *n*. For example, the equation

ν^{3} - *a*_{2}*v*^{2} + (*a*_{1}*a*_{3} - 4*α*_{4}) ν - (α\α_{Λ} - *Aα*_{2} α_{Λ} + *a*]) = 0

is one of the cubic resolvents of the fourth-degree equation

(1) *x*^{4} + *a*_{1}*x*^{3} + *a*_{2}*x*^{2} + *a*_{3}*x* + *a*_{4} = 0

If ν_{1}, *v*_{2}, and *v*_{3} are the roots of the resolvent equation, the roots *x*_{1}, *x*_{2}, *x*_{3}, and *x*_{4} of equation (1) can be found by solving the quadratic equations σ^{2} - *v _{k}* σ +

*a*

_{4}= 0,

*k*= 1, 2, 3. Thus, if

*x*and

_{n}*n*are the roots of these quadratic equations,

_{k}*x*

_{1}

*x*

_{2}=

*x*

_{3}

*x*

_{4}=

*n, x*

_{1}

*x*

_{3}=

*x*

_{2}

*x*

_{2}

*x*

_{4}=

*x*

_{2}

*x*

_{2}

*x*

_{4}=

*n*

_{3}, =

*ξ*

_{1}

*x*

_{2}/

*n*

_{3,}and so on. A Galois resolvent of the equation

*f*(

*x*) is an algebraic equation

*g*(

*x*) = 0 irreducible over a given field (

*see*GALOIS THEORY) such that when one of its roots is adjoined to the field, there results a field containing all the roots of the equation

*f*(

*x*) = 0.

The term “resolvent” is used in a somewhat different sense in what is known as the Hilbert-Chebotarev resolvent problem.

In the theory of integral equations, the resolvent of the equation

is a function Γ (*s, t;* λ) of the variables *s* and *t* and the parameter λ such that the solution of equation (2) can be represented in the form

provided that λ is not an eigenvalue of equation (2). For example, the resolvent of the kernel *K*(*s, t*) = *S* + *t* is the function

In the theory of linear operators, the resolvent of the operator *A* is the family of operators *R*_{λ} = (*A* - λ *E*)^{-1} where the complex parameter λ takes on any values outside the spectrum of *A*.