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

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Page 915

Hilbert space H and let E

Hilbert space H and let E

**denote**its resolution of the identity . Then there exists a sequence { x } C H such that H = -1 Ø ( ?; ) , where a Lil H x H ( x ; ) = sp { / ( T ) x ; l / e C ( o ( T ) ) } , and a decreasing sequence { en } ...Page 1126

0 0 a 0 of the closed set C ; we shall

0 0 a 0 of the closed set C ; we shall

**denote**this subspace of L2 [ 0 , 1 ] by the symbol L , ( C ) . Since each projection in the spectral resolution of T and hence each continuous function of T is a strong limit of linear combinations ...Page 1486

In the next few paragraphs t

In the next few paragraphs t

**denotes**a formally self adjoint formal differential operator of order n , defined on the interval R = { - 00 < t < +00 } . Let t have the form n i τ = Σα , ( 1 ) Ža , ( e ) ( ) dt j = and suppose that all ...### 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