## Linear Operators: Spectral theory |

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Page 1158

Let U be a

consider the

Since K is compact, there exist a finite number of points ml, . . ., mk such that the ...

Let U be a

**neighborhood**of 0 e R with compact closure, and for each n\ in Kconsider the

**neighborhood**N im1, U, — I = |me R\ \[x, m] — [x, toJ| < — ,xe U) □Since K is compact, there exist a finite number of points ml, . . ., mk such that the ...

Page 1678

Proof. Let $ be a second function in CJj°(7) such that xp(x) = 1 for x in a

vanishes in a

. Hence ...

Proof. Let $ be a second function in CJj°(7) such that xp(x) = 1 for x in a

**neighborhood**of Kv Then yxp—yxp vanishes in a**neighborhood**oi K n C(F), andvanishes in a

**neighborhood**of C(F)—K since <p vanishes in the complement of K. Hence ...

Page 1734

Let i7iC/0 be a bounded

that there exists a mapping q> of U1 onto the unit spherical

the origin such that (i) <p is one-to-one, is infinitely often differentiable, and qr1 is

...

Let i7iC/0 be a bounded

**neighborhood**of q chosen so small that fiU1 Q E, and sothat there exists a mapping q> of U1 onto the unit spherical

**neighborhood**V ofthe origin such that (i) <p is one-to-one, is infinitely often differentiable, and qr1 is

...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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### Common terms and phrases

Acad adjoint extension adjoint operator algebra Amer analytic 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 q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero