Linear Operators: Spectral theory |
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Page 890
for which 2 , ed , then the function E is a resolution of the identity for T and the operational calculus is given by the formula ( vi ) ( T ) = S , 7 , ( 2 ) E ( da ) , f where the integral is defined as the finite sum X - 11 ( 2 ...
for which 2 , ed , then the function E is a resolution of the identity for T and the operational calculus is given by the formula ( vi ) ( T ) = S , 7 , ( 2 ) E ( da ) , f where the integral is defined as the finite sum X - 11 ( 2 ...
Page 1089
This formula may be written achp max 1 , ... , ( un ( T ) ) 2 min 17912 19 = 1 ( 0,8 ) = ... ( Pn ) = 0 Since 1912 ( T9 , T9 ) = ( T * T9,9 ) , we see our lemma to be a special case of the “ minimax formula " for the eigenvalues of a ...
This formula may be written achp max 1 , ... , ( un ( T ) ) 2 min 17912 19 = 1 ( 0,8 ) = ... ( Pn ) = 0 Since 1912 ( T9 , T9 ) = ( T * T9,9 ) , we see our lemma to be a special case of the “ minimax formula " for the eigenvalues of a ...
Page 1363
basis for this formula is found in Theorem XII.2.10 which asserts that the projection in the resolution of the identity for T corresponding to ( 27 , 12 ) may be calculated from the resolvent by the formula 1 E ( ( 17 , 12 ) ) } = lim ...
basis for this formula is found in Theorem XII.2.10 which asserts that the projection in the resolution of the identity for T corresponding to ( 27 , 12 ) may be calculated from the resolvent by the formula 1 E ( ( 17 , 12 ) ) } = lim ...
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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