## Linear Operators: Spectral theory |

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Page 1037

product clearly converges to zero for A = A, it is readily seen that the function pa(T

) is

0, then pa(T) is continuous in T relative to the Hilbert-Schmidt norm in HS.

product clearly converges to zero for A = A, it is readily seen that the function pa(T

) is

**analytic**for A # 0 and vanishes only for A in a (T). It remains to show that if A #0, then pa(T) is continuous in T relative to the Hilbert-Schmidt norm in HS.

Page 1040

yı (A) is

Å; T)*y vanishes which will prove that y(A) is

that y(A) can only fail to be

) ...

yı (A) is

**analytic**even at A = Am. It will now be shown that ye(A) = AWE(7.3 T)*R(Å; T)*y vanishes which will prove that y(A) is

**analytic**at all the points A = Am, sothat y(A) can only fail to be

**analytic**at the point A = 0. To show this, note that (y,(A) ...

Page 1102

The determinant det(I+2T,) is an

To operates in finite-dimensional space, and hence more generally if T, has a

finite-dimensional range. Thus, since a bounded convergent sequence of

The determinant det(I+2T,) is an

**analytic**(and even a polynomial) function of z, ifTo operates in finite-dimensional space, and hence more generally if T, has a

finite-dimensional range. Thus, since a bounded convergent sequence of

**analytic**...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 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 |

### Common terms and phrases

additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense 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 measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero