## Linear Operators, Volume 2 |

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Page 1223

How are we to choose its

How are we to choose its

**domain**? A natural first guess is to choose as**domain**the collection D , of all functions with one continuous derivative . If f and g are any two such functions , we have ( iDf , g ) = So it ' ( ) g ( t ) dt ...Page 1248

The subspace M is called the initial

The subspace M is called the initial

**domain**of P and PM ( = P $ ) is called the final**domain**of P. 5 LEMMA . ... In this case PP * is also a projection and the ranges of P * P and PP * are the initial and final**domains**, respectively ...Page 1249

Thus PP * is a projection whose range is N = PM , the final

Thus PP * is a projection whose range is N = PM , the final

**domain**of P. To complete the proof it will suffice to show that P * P is a projection if P is a partial isometry . Let X , v EM , the initial**domain**of P. Then the identity \ x ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero