## Linear Operators: Spectral theory |

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Page 1092

Let Q be a finite-dimensional space including both the

of To; suppose that the dimension of 3 is d. Then, plainly, & is invariant under T

and T*, and, since (T&", w) = (3*, Toa.) = 0 for all a, we have T&H = 0 and similarly

...

Let Q be a finite-dimensional space including both the

**range**of T and the**range**of To; suppose that the dimension of 3 is d. Then, plainly, & is invariant under T

and T*, and, since (T&", w) = (3*, Toa.) = 0 for all a, we have T&H = 0 and similarly

...

Page 1134

Then, retracing the steps of the above argument, we can conclude that (I–EA)TEA

= 0 for each A in C. Hence T leaves the

and the set 3 of projections EA, A e C, subdiagonalizes T. To prove the second ...

Then, retracing the steps of the above argument, we can conclude that (I–EA)TEA

= 0 for each A in C. Hence T leaves the

**range**of each projection EA invariant,and the set 3 of projections EA, A e C, subdiagonalizes T. To prove the second ...

Page 1137

In this

< p < 00, and if T is a compact quasinilpotent operator whose anti-Hermitian part

belongs to the class C, then T itself belongs to the class C,. We will prepare for ...

In this

**range**, the following theorem states exactly such a result. 9 THEOREM. If 1< p < 00, and if T is a compact quasinilpotent operator whose anti-Hermitian part

belongs to the class C, then T itself belongs to the class C,. We will prepare for ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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