Resolvent

resolvent [ri′zäl·vənt]

(mathematics)

For a linear operatorTon a Banach space, the function, defined on the complement of the spectrum ofTgiven by (T- λI)^-1for each λ in this complement, whereIis the identity operator; this enables a study ofTrelative to its eigenvalues.

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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

ν³ - a₂v² + (a₁a₃ - 4α₄) ν - (α\α_Λ - Aα₂ α_Λ + a]) = 0

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

(1) x⁴ + a₁x³ + a₂x² + a₃x + a₄ = 0

If ν₁, v₂, and v₃ are the roots of the resolvent equation, the roots x₁, x₂, x₃, and x₄ of equation (1) can be found by solving the quadratic equations σ² - v_k σ + a₄ = 0, k = 1, 2, 3. Thus, if x_n and n_k are the roots of these quadratic equations, x₁x₂= x₃x₄ = n, x₁x₃ = x₂x₂x₄ = x₂x₂x₄ = n₃, = ξ₁x₂/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 (seeGALOIS 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.