## Linear Operators: Spectral theory |

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Page 1017

calculate the trace of A relative to the

cozov, j=1 and so, CAC-1 y, -2", $/; . From this it follows that the trace of CAC-4,

calculated relative to the

proved ...

calculate the trace of A relative to the

**basis**yi, ..., ya. Note that acou, -4-, -2°,-,-cozov, j=1 and so, CAC-1 y, -2", $/; . From this it follows that the trace of CAC-4,

calculated relative to the

**basis**{yi, ..., y,}, is X-1 air. By what has already beenproved ...

Page 1029

Then, since S is necessarily invariant under T, there exists by the inductive

hypothesis, an orthonormal

a, be orthogonal to S and have norm one so that {ri, ..., a,} is an orthonormal

for ...

Then, since S is necessarily invariant under T, there exists by the inductive

hypothesis, an orthonormal

**basis**{ri,..., r, 1} for S with ((T-21)r, w,) = 0 for j > i. Leta, be orthogonal to S and have norm one so that {ri, ..., a,} is an orthonormal

**basis**for ...

Page 1344

It is readily seen, by a similar argument, that p,(A) depends continuously on A. It

then follows easily from Corollary X.7.8 that E,(A) = E(M(4); p,(A)) depends

continuously on A, i = 1,..., k. Let v1,..., v, be an orthonormal

E ...

It is readily seen, by a similar argument, that p,(A) depends continuously on A. It

then follows easily from Corollary X.7.8 that E,(A) = E(M(4); p,(A)) depends

continuously on A, i = 1,..., k. Let v1,..., v, be an orthonormal

**basis**for E" such thatE ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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