## Linear Operators: Spectral operators |

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Page 895

The

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

...

The

**Spectral Theorem**for Bounded Normal Operators Before proving that abounded 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 I may now be applied to the restriction T. of T to the space XY, to yield a

regular positive measure u, vanishing on the complement of g(T ... We have seen

that Theorem 3 is a consequence of the

Theorem I may now be applied to the restriction T. of T to the space XY, to yield a

regular positive measure u, vanishing on the complement of g(T ... We have seen

that Theorem 3 is a consequence of the

**spectral theorem**for normal operators.Page 927

The

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 proofs of the spectral ...

The

**spectral theorem**. The**spectral theorem**for bounded self adjoint operators inHilbert 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 proofs of the spectral ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero