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

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Page 893

Let E be a

Let E be a

**bounded**self adjoint spectral measure in Hilbert space defined on a field of subsets of a set S. Then the map † T ( ) defined by the equation T ( 4 ) = st ( 8 ) E ( ds ) , fe B ( S , E ) , is a continuous * -homomorphic map ...Page 900

1 and thus there is a

1 and thus there is a

**bounded**function to on S with f ( s ) = to ( s ) except for s in a set having E measure zero . If f is E - measurable then to is a**bounded**E - measurable function , i.e. , an element of the B * -algebra BS , E ) .Page 1240

Semi -

Semi -

**bounded**Symmetric Operators In this section we study the self adjoint extensions of those operators in a class of symmetric operators which arise frequently from the boundary value problems of mathematical physics .### 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