## Linear Operators: Spectral theory |

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Page 877

Then an

Consequently the spectrum of y as an

as an

Then an

**element**y in ?) has an inverse in X if and only if it has an inverse in 9J.Consequently the spectrum of y as an

**element**of 9) is the same as its spectrumas an

**element**of X. Proof. If y~l exists as an**element**of 9J then, since X and 9J ...Page 878

Clearly the requirement that x and g(/x) = fi be corresponding

determines the *-iso- morphism uniquely and we are thus led to the following

definition. 12 Definition. Let x be an

f e C(a(x)).

Clearly the requirement that x and g(/x) = fi be corresponding

**elements**determines the *-iso- morphism uniquely and we are thus led to the following

definition. 12 Definition. Let x be an

**element**of a commutative ^-algebra and letf e C(a(x)).

Page 1339

An

equivalence classes of

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

above is ...

An

**element**F of will be said to be a {fi^-null function if |F| = 0. The set of allequivalence classes of

**elements**of ^({^.j}) modulo {u^-null functions will bedenoted by L2({//(j}). We observe that by Lemma 7, the integrand in the integral

above is ...

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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