## Linear Operators, Volume 2 |

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Page 1224

( b ) If T , is symmetric then every symmetric

( b ) If T , is symmetric then every symmetric

**extension**T , of T ,, and , in particular , every self adjoint**extension**of Tı , satisfies T , CT , CT CT * Proof . If T , CT , and y e D ( 1 * ) , then ( x , T * y ) = ( Tox , y ) ( T ...Page 1239

Conversely , let T , be a self adjoint

Conversely , let T , be a self adjoint

**extension**of T. Then by Lemma 26 , T is the restriction of T * to a subspace W of D ( T * ) determined by a symmetric family of linearly independent boundary conditions B1 ( x ) = 0 , i = 1 , ...Page 1270

**Extensions**of symmetric operators . The problem of determining whether a given symmetric operator has a self adjoint**extension**is of crucial importance in determining whether the spectral theorem may be employed .### 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