## Linear Operators: Spectral theory |

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Page 1215

Thus $sf(s)Wa(s> X)v{ds) exists in the mean square sense and equals (Uf)a(X),

proving (c). Q.E.D. Using the notation of the

that, by Lemma 9, J"_"n (Uj)a{X)Wa{-,X)na{dX) = E([-n, n])F(T)a -> F(T)a = U-*F = /

„.

Thus $sf(s)Wa(s> X)v{ds) exists in the mean square sense and equals (Uf)a(X),

proving (c). Q.E.D. Using the notation of the

**preceding**proof we let F = (Uf)a sothat, by Lemma 9, J"_"n (Uj)a{X)Wa{-,X)na{dX) = E([-n, n])F(T)a -> F(T)a = U-*F = /

„.

Page 1480

Under the hypotheses and with the notation of the

defined by at most one boundary condition, which is a boundary condition at an

end point a of the interval I. Let X < Xq, and let a(t, X) be a solution of the equation

to ...

Under the hypotheses and with the notation of the

**preceding**theorem, T isdefined by at most one boundary condition, which is a boundary condition at an

end point a of the interval I. Let X < Xq, and let a(t, X) be a solution of the equation

to ...

Page 1771

... x) = 0, t>0, xeZ, and such that lim \u(t, •)—/(•)! = 0- «-.o Proof. Statement (i)

follows from the

statement (ii) of the

of ...

... x) = 0, t>0, xeZ, and such that lim \u(t, •)—/(•)! = 0- «-.o Proof. Statement (i)

follows from the

**preceding**theorem and Theorem 6.23. Statement (ii) follows fromstatement (ii) of the

**preceding**theorem, since a function satisfying the hypothesesof ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic 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 q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero