Linear Operators: Spectral theory |
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Page 895
... Spectral Theorem for Bounded Normal Operators Before proving that a bounded normal operator in Hilbert space has a resolution of the identity , we will prove the following more general theorem which will be used frequently when its ...
... Spectral Theorem for Bounded Normal Operators Before proving that a bounded normal operator in Hilbert space has a resolution of the identity , we will prove the following more general theorem which will be used frequently when its ...
Page 911
... theorem . α α α α 3 THEOREM . Every Hilbert space admits a spectral representation relative to an arbitrary bounded normal operator defined in it . = We have seen that Theorem 3 is a consequence of the spectral theorem for normal ...
... theorem . α α α α 3 THEOREM . Every Hilbert space admits a spectral representation relative to an arbitrary bounded normal operator defined in it . = We have seen that Theorem 3 is a consequence of the spectral theorem for normal ...
Page 927
... spectral theorem . The spectral theorem for bounded self ad- joint operators in Hilbert space is due to Hilbert [ 1 ; IV ] . The reader should also see the proofs of F. Riesz [ 20 , 6 ] which are quite modern in spirit . Many other ...
... spectral theorem . The spectral theorem for bounded self ad- joint operators in Hilbert space is due to Hilbert [ 1 ; IV ] . The reader should also see the proofs of F. Riesz [ 20 , 6 ] which are quite modern in spirit . Many other ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
Copyright | |
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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