## Linear operators. 2. Spectral theory : self adjoint operators in Hilbert Space |

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Page 877

Then an

Then an

**element**y in Y has an inverse in X if and only if it has an inverse in y . Consequently the spectrum of y as an**element**of y is the same as its spectrum as an**element**of X. Proof . If y - l exists as an**element**of Y then , since ...Page 878

Clearly the requirement that x and g ( u ) = u be corresponding

Clearly the requirement that x and g ( u ) = u be corresponding

**elements**determines the * -isomorphism uniquely and we are thus led to the following definition . a n nm 1 12 DEFINITION . Let x be an**element**of a commutative B * -algebra ...Page 1339

An

An

**element**F of L ( { M i ; } ) will be said to be a { Mi ; } - null function if [ F ] = 0. The set of all equivalence classes of**elements**of L } ( { uis } ) modulo { ui ; } - null functions will be denoted by L2 ( { uis } ) .### What people are saying - Write a review

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