## Linear Operators: Spectral theory |

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Page 961

FWO) – 37-7). Since the operation T(f) of convolution by f commutes with E(e)

and since, as we have seen in the

...

**PRoof**. If we write p for p(e), then (s. v) = s.sosode = s.st-ood, = s.so-TVG) is - (FWO) – 37-7). Since the operation T(f) of convolution by f commutes with E(e)

and since, as we have seen in the

**proof**of Lemma 6, E(e)y = p, it follows from the...

Page 1012

from which it follows that ||T–Tall se for m > m(e) and completes the

is a B-space under the Hilbert-Schmidt norm. Finally, let T be in HS and let B be

any bounded linear operator in H. Then pro-x.or.'s box, T. -poro. |TB|| = ||(TB)"|| ...

from which it follows that ||T–Tall se for m > m(e) and completes the

**proof**that HSis a B-space under the Hilbert-Schmidt norm. Finally, let T be in HS and let B be

any bounded linear operator in H. Then pro-x.or.'s box, T. -poro. |TB|| = ||(TB)"|| ...

Page 1179

sends a scalar-valued function with the Fourier transform f($) into the vector-

valued function whose nth component has the Fourier transform f,(5) defined by (

65) ...

**PRoof**. We saw in the course of proving Theorem 25 that the mapping .4% whichsends a scalar-valued function with the Fourier transform f($) into the vector-

valued function whose nth component has the Fourier transform f,(5) defined by (

65) ...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

38 other sections not shown

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex 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 matrix 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