## Linear Operators: Spectral theory |

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Page 1223

How are we to choose its

the collection 2)j of all functions with one continuous derivative. If / and g are any

two such functions, we have (iDf,g) = ^if'(l)g~{t)dt = (0'/(<)W+i(/(iteli)-/(0OT) = (/ ...

How are we to choose its

**domain**? A natural first guess is to choose as**domain**the collection 2)j of all functions with one continuous derivative. If / and g are any

two such functions, we have (iDf,g) = ^if'(l)g~{t)dt = (0'/(<)W+i(/(iteli)-/(0OT) = (/ ...

Page 1249

Thus PP* is a projection whose range is 31 = P3R, the final

complete the proof it will suffice to show that P*P is a projection if P is a partial

isometry. Let x, v e3Jl, the initial

shows ...

Thus PP* is a projection whose range is 31 = P3R, the final

**domain**of P. Tocomplete the proof it will suffice to show that P*P is a projection if P is a partial

isometry. Let x, v e3Jl, the initial

**domain**of P. Then the identity \x+v\2 = \Px+Pv\2shows ...

Page 1669

Let Ix be a

mapping of I1 into I2 such that (a) M~lC is a compact subset of Zj whenever C is a

compact subset of I2; (b) (M(-))JeC~(/1), ; = l,...,nt. Then (i) for each <p in C°°(I2), <

poM ...

Let Ix be a

**domain**in En\ and let I2 be a**domain**in En*. Let M : It -> I2 be amapping of I1 into I2 such that (a) M~lC is a compact subset of Zj whenever C is a

compact subset of I2; (b) (M(-))JeC~(/1), ; = l,...,nt. Then (i) for each <p in C°°(I2), <

poM ...

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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