## Linear Operators: Spectral operators |

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Page 968

Verification that the

to the reader. If hi e N(h, K, s) and h, e N(ho, K, e) then h, he e N(hho, K*, *) so

that multiplication is continuous. If hi e N(h, K, e) then hio e N(h-", K, e), so the ...

Verification that the

**neighborhoods**N(h, K, e) are a base for a topology will be leftto the reader. If hi e N(h, K, s) and h, e N(ho, K, e) then h, he e N(hho, K*, *) so

that multiplication is continuous. If hi e N(h, K, e) then hio e N(h-", K, e), so the ...

Page 1303

Clearly B(f) = 0 for those f which vanish in a

boundary value for r at a. To prove the converse, let B be a boundary value at a.

Choose a function h in C*(I) which is identically equal to one in a

of a ...

Clearly B(f) = 0 for those f which vanish in a

**neighborhood**of a. Thus B is aboundary value for r at a. To prove the converse, let B be a boundary value at a.

Choose a function h in C*(I) which is identically equal to one in a

**neighborhood**of a ...

Page 1733

Q.E.D. Lemma 18 enables us to use the method of proof of Theorem 2 in the

out in the next two lemmas. 19 LEMMA. Let G be an elliptic formal partial

differential ...

Q.E.D. Lemma 18 enables us to use the method of proof of Theorem 2 in the

**neighborhood**of the boundary of a domain with smooth boundary. This is carriedout in the next two lemmas. 19 LEMMA. Let G be an elliptic formal partial

differential ...

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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