## Linear Operators: Spectral theory |

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Page 1040

91(2) is

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

y(A) can only fail to be

91(2) is

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

**analytic**at all the points A = A, , so thaty(A) can only fail to be

**analytic**at the point A = 0. To show this, note that (y,(2), ...Page 1102

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

T, 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-I-2T,) is an

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

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

**analytic**...Page 1364

It follows by induction that we can construct the required functionals p1, ..., py in

St. We now select a neighborhood G(Wo) of Åo such that the

(2)} has a non-vanishing determinant for A e G(Zo). It follows easily that {q, Q,(2)}

...

It follows by induction that we can construct the required functionals p1, ..., py in

St. We now select a neighborhood G(Wo) of Åo such that the

**analytic**matrix {q, Q,(2)} has a non-vanishing determinant for A e G(Zo). It follows easily that {q, Q,(2)}

...

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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