## Linear Operators: Spectral theory |

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Page 915

Hilbert space Sy and let E

sequence {a,) C. § such that § = X.1 $5(x,), where $3 (r) = SE {f(T)a,fe C(a(T))},

and a decreasing sequence {e,} of Borel sets such that (E(e)w, w,) = (E(ee,)ri, arı),

...

Hilbert space Sy and let E

**denote**its resolution of the identity. Then there erists asequence {a,) C. § such that § = X.1 $5(x,), where $3 (r) = SE {f(T)a,fe C(a(T))},

and a decreasing sequence {e,} of Borel sets such that (E(e)w, w,) = (E(ee,)ri, arı),

...

Page 1126

of the closed set C; we shall

Since each projection in the spectral resolution of T and hence each continuous

function of T is a strong limit of linear combinations of the projections E, ...

of the closed set C; we shall

**denote**this subspace of L2[0, 1] by the symbol L.(C ).Since each projection in the spectral resolution of T and hence each continuous

function of T is a strong limit of linear combinations of the projections E, ...

Page 1486

In the next few paragraphs t

operator of order n, defined on the interval R = {– o – t < -i- oc). Let t have the form

n d j -3.0 (). and suppose that all the coefficients a, are periodic and have the

same ...

In the next few paragraphs t

**denotes**a formally self adjoint formal differentialoperator of order n, defined on the interval R = {– o – t < -i- oc). Let t have the form

n d j -3.0 (). and suppose that all the coefficients a, are periodic and have the

same ...

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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