## Linear operators: Spectral operators |

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Page 2128

This result provides a key to a rapid proof of the spectral theorem for normal

in

This result provides a key to a rapid proof of the spectral theorem for normal

**operators**. There are a number of other aspects of the spectral theory of**operators**in

**Hilbert**space, such as the notion of the numerical range,**Hilbert**-**Schmidt**...Page 2488

(e) Prove that if A is an

and \\w\\H < oo, then f + "\AeitHw\2 dtŁ2ir \\A |j| |M& . (Hint: Consider .4 self adjoint

, and expand in the eigenvectors of A.) 17 (a) Let / be a continuous function of ...

(e) Prove that if A is an

**operator**of the**Hilbert**-**Schmidt**class, while w e Łac (//)and \\w\\H < oo, then f + "\AeitHw\2 dtŁ2ir \\A |j| |M& . (Hint: Consider .4 self adjoint

, and expand in the eigenvectors of A.) 17 (a) Let / be a continuous function of ...

Page 2489

Utw exists and \{W+ - Us)w\2 = C(V exp(-ixH1)w,W + expi-ixHJw) dx. •'j (Hint: Use

Theorem 4.9.) (d) (Rosenblum's Inequality) Show that there exist

of

Utw exists and \{W+ - Us)w\2 = C(V exp(-ixH1)w,W + expi-ixHJw) dx. •'j (Hint: Use

Theorem 4.9.) (d) (Rosenblum's Inequality) Show that there exist

**operators**A, Bof

**Hilbert**-**Schmidt**class depending only on V and p such that if w 6 Łoc (Hj) and ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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adjoint operator algebra of projections Amer analytic arbitrary asymptotic B-space Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote dense differential operator disjoint Doklady Akad domain eigenvalues elements equation exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix measurable functions multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proof properties prove quasi-nilpotent restriction Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset Suppose trace class type spectral operator unbounded uniformly bounded unique vector weakly complete zero