## Linear Operators: Spectral theory |

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Page 1250

Finally we show that the decomposition T = PA of the theorem is

A is

Further the extension of P by continuity from R ( A ) to R ( A ) is

Finally we show that the decomposition T = PA of the theorem is

**unique**. ... SinceA is

**unique**, P is**uniquely**determined on R ( A ) by the equation of P ( Ax ) = Tx .Further the extension of P by continuity from R ( A ) to R ( A ) is

**unique**. Since P ...Page 1283

Thus , equation ( e ' ) has the

H = ( - 1 ) P H . j = 0 Since all the terms in equation ( e ) but the first are absolutely

continuous , it follows that F is absolutely continuous . Thus Theorem 1 is ...

Thus , equation ( e ' ) has the

**unique**solution ( cf . Lemma VII . 3 . 4 ) F = ( 1 + 0 ) -H = ( - 1 ) P H . j = 0 Since all the terms in equation ( e ) but the first are absolutely

continuous , it follows that F is absolutely continuous . Thus Theorem 1 is ...

Page 1383

With boundary conditions A and C , the

boundary condition T30 = ho is sin vīt . With boundary conditions A , the

eigenvalues are consequently to be determined from the equation sin vă = 0 .

With boundary conditions A and C , the

**unique**solution of tzo = lo satisfying theboundary condition T30 = ho is sin vīt . With boundary conditions A , the

eigenvalues are consequently to be determined from the equation sin vă = 0 .

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

46 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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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 consider 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 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