Linear Operators: Spectral theory |
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Page 861
If t ; exists in B ( X ) , then T T.T. Tz ' T . [ ( T ; ' Y ) 2 ] = yz , ( T ' y ) z = T ?? ( yz ) , : Tole ... An element æ in a B - algebra X is said to be regular in case r - 1 exists in X. It is singular if it is not regular .
If t ; exists in B ( X ) , then T T.T. Tz ' T . [ ( T ; ' Y ) 2 ] = yz , ( T ' y ) z = T ?? ( yz ) , : Tole ... An element æ in a B - algebra X is said to be regular in case r - 1 exists in X. It is singular if it is not regular .
Page 1057
By Lemma 2 , the integral 0 ( tu ) exists if 0 ( u ) exists and t > 0 ; and the integral 0 ( Vu ) exists and equals S. eiyu dy piyu dy n E - 0 € 5 \ HSR \ y \ " 1 S2 ( x ) ei ( x , Vu ) dx PS . 2 ( Vy ) ei ( yu ) dy En form En ly " En ...
By Lemma 2 , the integral 0 ( tu ) exists if 0 ( u ) exists and t > 0 ; and the integral 0 ( Vu ) exists and equals S. eiyu dy piyu dy n E - 0 € 5 \ HSR \ y \ " 1 S2 ( x ) ei ( x , Vu ) dx PS . 2 ( Vy ) ei ( yu ) dy En form En ly " En ...
Page 1733
Then , if f is in HP ( I ) and of is in H ( m ) ( I ) , there exists a neighborhood V of E , such that the restriction of f to VI belongs to H ( 2P + m ( VI ) . This lemma will be deduced from the following lemma : 20 LEMMA .
Then , if f is in HP ( I ) and of is in H ( m ) ( I ) , there exists a neighborhood V of E , such that the restriction of f to VI belongs to H ( 2P + m ( VI ) . This lemma will be deduced from the following lemma : 20 LEMMA .
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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