## Linear Operators: Spectral operators |

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Page 1217

A spectral representation of a Hilbert space \) onto X.1. L2(u,) relative to a self

adjoint operator

The measure su is called the measure of the ordered representation. The sets e,

will ...

A spectral representation of a Hilbert space \) onto X.1. L2(u,) relative to a self

adjoint operator

**T**in \) is said to be an ordered representation of S) relative**to T**.The measure su is called the measure of the ordered representation. The sets e,

will ...

Page 1246

We may also regard A as a mapping from the dense subspace o(

space S). In this case A is still continuous, for |Aali = (Aw, Ar), - (A*r, r) = (Ar, w),

a e o (

Airl ...

We may also regard A as a mapping from the dense subspace o(

**T**) of S) into thespace S). In this case A is still continuous, for |Aali = (Aw, Ar), - (A*r, r) = (Ar, w),

a e o (

**T**), and, by the inequalities above, (Aar, r) s |Aar lar s |Arial, showing thatAirl ...

Page 1437

Then, since To(r) C Ti(r), it follows immediately from the preceding lemma that

Zoe G.(Ti(r)), so that by Definition 6.1, Žo e o (

be the closure in the Hilbert space Q(Ti(r)) of £(To(r)), and let

restriction ...

Then, since To(r) C Ti(r), it follows immediately from the preceding lemma that

Zoe G.(Ti(r)), so that by Definition 6.1, Žo e o (

**t**). Conversely, let Zoe G.(r). Let },be the closure in the Hilbert space Q(Ti(r)) of £(To(r)), and let

**To(t**) be therestriction ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### Common terms and phrases

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