Linear Operators: Spectral theory

additive algebra analytic assume B-algebra basis Borel set boundary values bounded called character clear closed closure commutative compact complex Consequently consider constant contains continuous function converges Corollary corresponding defined Definition denote dense determined differential operator eigenvalues element equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formula function function f given Hence Hilbert space Hilbert-Schmidt homomorphism ideal identity inequality integral interval isometric isomorphism Lemma limit linear Ly(R matrix measure multiplicity neighborhood norm normal operator obtained operator orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero