## Linear Operators: Spectral theory |

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Page 994

translation. Thus S, the conjugate-orthogonal complement in L^iR) of £, is also

closed under translation. Now [•, m0] is the only character in $ since, if tn ^ m0,

Lemma 3 shows that there is a

a ...

translation. Thus S, the conjugate-orthogonal complement in L^iR) of £, is also

closed under translation. Now [•, m0] is the only character in $ since, if tn ^ m0,

Lemma 3 shows that there is a

**function g**in LX(R) whose transform ^ vanishes ina ...

Page 1292

Then there exists a

there is a unique

boundary conditions: [**] g(fl) = g'(a) = . . . = g(»-i»(fl) = 0. By Green's formula and

by ...

Then there exists a

**function g**in //£(/) such that T*g = w. Indeed, by Corollary 1.3there is a unique

**function g**in C"(I) satisfying the equation r*g = w and theboundary conditions: [**] g(fl) = g'(a) = . . . = g(»-i»(fl) = 0. By Green's formula and

by ...

Page 1644

t^iPgitx) -+dJg{x) uniformly in x for each

/ is a Lebesgue-integrable

denote the integral of / over the set e. Let / be an open subset of En, x a formal ...

t^iPgitx) -+dJg{x) uniformly in x for each

**g**in Cg(£"), provided that \J\ ^ p. Q.E.D. If/ is a Lebesgue-integrable

**function**defined on a subset e of E", then jj(x)dx willdenote the integral of / over the set e. Let / be an open subset of En, x a formal ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

Copyright | |

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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