SOLVED:Prove or give a counterexample: every self-adjoint operator on V has a cube root. (An operator S ∈ℒ(V) is called a cube root of T ∈ℒ(V) if .S^3=T .)

Define Self-Adjoint Operator and prove that the sum of two Self... | Filo

SOLVED:(a) Suppose that the self-adjoint operator 𝒮 in Problem 3(b) is defined on a space of functions satisfying the conditions a1 X(a) a2 X^'(a)=0, b1 X(b) b2 X^'(b)=0 where a1 and a2 are not both zero and where the same is true of b1 and b2. Use Lagrange's identity, obtained in that problem, to show that (X, ℒ[Y])=(ℒ[X], Y) where these inner products are on the interval a<x<b and with weight function unity. (b) Let λm and λn denote distinct eigenvalues of a regular Sturm-Liouville problem,

Adjoint Operator (20 points) Derive the following properties of... | Filo

The formula (Af)(t):=tf(t) defines a continuous self-adjoint op... | Filo