Linear Operators: Spectral theory |
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Page 888
... spectral measure E defined on the family of spectral sets of T. This spectral measure is also related ( VII.3.20 ) to T by the equations ( iii ) E ( S ) T = TE ( 8 ) , o ( T ) = 8 δ = E ( 8 ; ) X is d1 . where & is an arbitrary spectral ...
... spectral measure E defined on the family of spectral sets of T. This spectral measure is also related ( VII.3.20 ) to T by the equations ( iii ) E ( S ) T = TE ( 8 ) , o ( T ) = 8 δ = E ( 8 ; ) X is d1 . where & is an arbitrary spectral ...
Page 889
... spectral measure satisfying ( iii ) is necessarily an open and closed subset of o ( T ) and thus a spectral set . However , in order to reduce the study of T to its study on invariant subspaces in which it has a smaller spectrum it is ...
... spectral measure satisfying ( iii ) is necessarily an open and closed subset of o ( T ) and thus a spectral set . However , in order to reduce the study of T to its study on invariant subspaces in which it has a smaller spectrum it is ...
Page 897
... spectral measure and , in particular , all of the projections E ( 8 ) commute . It follows then from ( iii ) that the projections E ( 8 ) also commute with T ( f ) and this completes the proof of the theorem . Q.E.D. 3 COROLLARY . The ...
... spectral measure and , in particular , all of the projections E ( 8 ) commute . It follows then from ( iii ) that the projections E ( 8 ) also commute with T ( f ) and this completes the proof of the theorem . Q.E.D. 3 COROLLARY . The ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero