## Linear Operators: Spectral theory |

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Page 1226

Part (a) follows immediately from

from part (a) and

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Part (a) follows immediately from

**Lemma**5(b), and part (b) follows immediatelyfrom part (a) and

**Lemma**5(c). Q.E.D. It follows from**Lemma**6(b) that anysymmetric operator with dense domain has a unique minimal closed symmetric

extension.

Page 1698

Using

so that in case (a) we have shown that focp\y is the limit in the norm of //<P,(F) of ...

Using

**Lemma**2.1, let y> be a function in CJj°(F) with ip(x) = 1 for x in K. Then, by**Lemmas**3.22 and 3.10, /oyj"1 = y>(f 0<Pil) — umm-.ooV'(m m the norm of //""(F),so that in case (a) we have shown that focp\y is the limit in the norm of //<P,(F) of ...

Page 1733

Q.E.D.

neighborhood of the boundary of a domain with smooth boundary. This is carried

out in the next two

differential ...

Q.E.D.

**Lemma**18 enables us to use the method of proof of Theorem 2 in theneighborhood of the boundary of a domain with smooth boundary. This is carried

out in the next two

**lemmas**. 19**Lemma**. Let a be an elliptic formal partialdifferential ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero