## Linear Operators: Spectral theory |

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Page 865

o,(r) and g(r) are equal. An

where eo is an idempotent with 0 A eo A e clearly has g(t) Co(r). The following

lemma shows that the opposite inclusion holds in case &o has the same unit as 3

.

o,(r) and g(r) are equal. An

**element**r in a B-subalgebra of the form \, = toeowhere eo is an idempotent with 0 A eo A e clearly has g(t) Co(r). The following

lemma shows that the opposite inclusion holds in case &o has the same unit as 3

.

Page 877

Then an

Consequently the spectrum of y as an

as an

Then an

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

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

**element**of 3. Proof. If yo-" exists as an**element**of 9) 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({u,}) modulo (u,)-null functions will bedenoted by L2({u,}). We observe that by Lemma 7, the integrand in the integral ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers 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 f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero