## Linear Operators: Spectral theory |

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Page 911

It is clear that we may regard So as the direct sum S = X $5, of the Hilbert

S5, (cf. Lemma IV.4.19). Theorem 1 may now be applied to the restriction T. of T

to the

It is clear that we may regard So as the direct sum S = X $5, of the Hilbert

**spaces**S5, (cf. Lemma IV.4.19). Theorem 1 may now be applied to the restriction T. of T

to the

**space**\}, to yield a regular**positive measure**u, vanishing on the ...Page 1173

Let (S, 2, u) be a

defined in E", homogeneous of order 0, smooth except at a = 0, and whose

surface integral over the surface of the unit sphere is zero. Let L,(L.) denote the ...

Let (S, 2, u) be a

**positive measure space**. Let Q(r) be a numerically-valued kerneldefined in E", homogeneous of order 0, smooth except at a = 0, and whose

surface integral over the surface of the unit sphere is zero. Let L,(L.) denote the ...

Page 1210

Let T be a self adjoint operator in the Hilbert space L2(S, X, v) where (S. 2, v) is a

on each set in an increasing sequence of sets of finite measure which covers S.

Let T be a self adjoint operator in the Hilbert space L2(S, X, v) where (S. 2, v) is a

**positive measure space**. Let every element in s].13 (T") be v-essentially boundedon each set in an increasing sequence of sets of finite measure which covers S.

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

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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