## Linear Operators: Spectral theory |

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Page 1669

Let M : It -> I2 be a

Zj whenever C is a compact subset of I2; (b) (M(-))JeC~(/1), ; = l,...,nt. Then (i) for

each <p in C°°(I2), <poM will denote the function y> in C^/j) defined, for x in Ilt by

...

Let M : It -> I2 be a

**mapping**of I1 into I2 such that (a) M~lC is a compact subset ofZj whenever C is a compact subset of I2; (b) (M(-))JeC~(/1), ; = l,...,nt. Then (i) for

each <p in C°°(I2), <poM will denote the function y> in C^/j) defined, for x in Ilt by

...

Page 1671

If F corresponds to the function /, we have (foitf-'Xy) = F(<poM) = j f(x)<p(M(x))dx =

j f{M~1(x))9(x)J(x)dx, J denoting the absolute value of the Jacobian determinant

of the

If F corresponds to the function /, we have (foitf-'Xy) = F(<poM) = j f(x)<p(M(x))dx =

j f{M~1(x))9(x)J(x)dx, J denoting the absolute value of the Jacobian determinant

of the

**mapping**x M~1(x); this follows by the standard theorem on change of ...Page 1736

The

Lemmas 3.22 and 3.23, and evidently

Definition 3.15 that it

//qP)(C).

The

**mapping**g -+ fg|C is a continuous**mapping**of //'"(e-1/) into i/(p,(C) byLemmas 3.22 and 3.23, and evidently

**maps**C£°(/) into C~(C). It follows fromDefinition 3.15 that it

**maps**J/J,"^1/) into Hl0p)(C). Thus, fE = ^foS~l)\C belongs to//qP)(C).

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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