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

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Page 1083

Consequently , the series o 4 ( 2 ) — 28 ( 2 ) 1 = 24 , 2n n n = 2 of the preceding

Consequently , the series o 4 ( 2 ) — 28 ( 2 ) 1 = 24 , 2n n n = 2 of the preceding

**exercise**converges in the Hilbert - Schmidt norm . 44 Let ( S , E , u ) be a positive measure space . Then an operator A in the Hilbert space L ( S ...Page 1086

converges for all 2 , for u Xu - almost all ( s , t ) , and D ( s , t ; 2 ) is the kernel which represents the operator D ( 2 ) -1d ( 2 ) 1 of the preceding

converges for all 2 , for u Xu - almost all ( s , t ) , and D ( s , t ; 2 ) is the kernel which represents the operator D ( 2 ) -1d ( 2 ) 1 of the preceding

**exercise**, in the sense of**Exercise**44 . Show , finally , that by choosing As ...Page 1087

( Hint : For ( d ) , use Weyl's inequality ,

( Hint : For ( d ) , use Weyl's inequality ,

**Exercise**30. ) E. Miscellaneous**Exercises**> > P2 2 50 ( Halberg ) Let ( S , E , u ) be a o - finite measure space . Let T , be a l - parameter family of bounded operators defined in a ...### 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