## Linear Operators: Spectral operators |

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Page 865

An

with 0 # eo A e clearly has go(r) Co(a). The following lemma shows that the

opposite inclusion holds in case 3.0 has the same unit as 3. 9 LEMMA. Let a be

an ...

An

**element**a in a B-subalgebra of the form to - eo?eo where eo is an idempotentwith 0 # eo A e clearly has go(r) Co(a). The following lemma shows that the

opposite inclusion holds in case 3.0 has the same unit as 3. 9 LEMMA. Let a be

an ...

Page 877

Then an

Consequently the spectrum of y as an

as an

...

Then an

**element**y in Q) has an inverse in 3: if and only if it has an inverse in Q).Consequently the spectrum of y as an

**element**of Q) is the same as its spectrumas an

**element**of 3. A o Proof. If yol exists as an**element**of Q) then, since 3 and 9)...

Page 1339

An

equivalence classes of

denoted by L2({u,}). We observe that by Lemma 7, the integrand in the integral ...

An

**element**F of L(u,H) will be said to be a {u, null function if F = 0. The set of allequivalence classes of

**elements**of L. (Kuo) modulo (u,3-mull functions will bedenoted by L2({u,}). We observe that by Lemma 7, the integrand in the integral ...

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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