Linear Operators: Spectral theory |
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Page 1247
If T is a positive self adjoint transformation , there is a unique positive self adjoint transformation A such that A2 = T. Proof . By Lemma 2 , ( T ) C 0 , 0 ) and , by Theorem 2.6 ( d ) , the positive function f ( 2 ) = 24 on o ( T ) ...
If T is a positive self adjoint transformation , there is a unique positive self adjoint transformation A such that A2 = T. Proof . By Lemma 2 , ( T ) C 0 , 0 ) and , by Theorem 2.6 ( d ) , the positive function f ( 2 ) = 24 on o ( T ) ...
Page 1250
Finally we show that the decomposition T = PA of the theorem is unique . ... Since A is unique , P is uniquely determined on R ( A ) by the equation of P ( Ax ) = Tx . Further the extension of P by continuity from R ( A ) to R ( A ) is ...
Finally we show that the decomposition T = PA of the theorem is unique . ... Since A is unique , P is uniquely determined on R ( A ) by the equation of P ( Ax ) = Tx . Further the extension of P by continuity from R ( A ) to R ( A ) is ...
Page 1283
Thus , equation ( e ' ) has the unique solution ( cf. Lemma VII.3.4 ) . F = ( 1 + 0 ) -H Σ Σ ( -1 ) Φ ' Η . j = 0 Since all the terms in equation ( e ) but the first are absolutely continuous , it follows that F is absolutely continuous ...
Thus , equation ( e ' ) has the unique solution ( cf. Lemma VII.3.4 ) . F = ( 1 + 0 ) -H Σ Σ ( -1 ) Φ ' Η . j = 0 Since all the terms in equation ( e ) but the first are absolutely continuous , it follows that F is absolutely continuous ...
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Contents
8 | 876 |
859 | 885 |
extensive presentation of applications of the spectral theorem | 911 |
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additive adjoint adjoint operator algebra analytic assume 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 domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero