Linear Operators: Spectral theory |
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Page 897
It follows then from ( iii ) that the projections E ( d ) also commute with T ( / ) and this completes the proof of the theorem . Q.E.D. 3 COROLLARY . The spectral measure is countably additive in the strong operator topology . Proof .
It follows then from ( iii ) that the projections E ( d ) also commute with T ( / ) and this completes the proof of the theorem . Q.E.D. 3 COROLLARY . The spectral measure is countably additive in the strong operator topology . Proof .
Page 922
n n n n n n n n n n n n = n n n n topology , i.e. , Tnx → Tx for every x in the space upon which the operators T , T1 , T2 , ... , are defined , 1 LEMMA . Let S , T , S. , Tn , n 21 be bounded linear operators in Hilbert space with S ...
n n n n n n n n n n n n = n n n n topology , i.e. , Tnx → Tx for every x in the space upon which the operators T , T1 , T2 , ... , are defined , 1 LEMMA . Let S , T , S. , Tn , n 21 be bounded linear operators in Hilbert space with S ...
Page 1921
F ( 1550 ) Subadditive function , definition , ( 618 ) Subbase for a topology , 1.4.6 ( 10 ) criterion for , 1.4.8 ( 11 ) Subspace , of a linear space , ( 36 ) . ( See also Manifold ) Summability , of Fourier series , IV.14.34-51 ...
F ( 1550 ) Subadditive function , definition , ( 618 ) Subbase for a topology , 1.4.6 ( 10 ) criterion for , 1.4.8 ( 11 ) Subspace , of a linear space , ( 36 ) . ( See also Manifold ) Summability , of Fourier series , IV.14.34-51 ...
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Contents
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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