## Linear Operators: Spectral operators |

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Page 978

nondiscrete locally compact Abelian group and integration will always be

performed with respect to a Haar measure on the group. It was observed in

Corollary 5.2 that ...

**Closure**Theorems As in the preceding section the letter R will stand for anondiscrete locally compact Abelian group and integration will always be

performed with respect to a Haar measure on the group. It was observed in

Corollary 5.2 that ...

Page 993

Now let V, be an arbitrary open subset of R with compact

from what has just been demonstrated that zy, a zvuv, - gy, i.e., zy is independent

of V. Q.E.D. 16 THEOREM. If the bounded measurable function p has its ...

Now let V, be an arbitrary open subset of R with compact

**closure**. Then it followsfrom what has just been demonstrated that zy, a zvuv, - gy, i.e., zy is independent

of V. Q.E.D. 16 THEOREM. If the bounded measurable function p has its ...

Page 1226

The minimal closed symmetric extension of a symmetric operator T with dense

domain is called its

restriction of To to the

T ...

The minimal closed symmetric extension of a symmetric operator T with dense

domain is called its

**closure**, and written T. 8 LEMMA. (a) The**closure**T of T is therestriction of To to the

**closure**of Q(T) in the Hilbert space 3 (T"). (b) The operatorT ...

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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