## Linear Operators: Spectral theory |

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Page 1257

5 If T is a densely defined symmetric transformation, [+] V = (T-iH)(T-Hil)-1 is an

following: The operator T is closed if and only if V is closed. The operator I–V is ...

5 If T is a densely defined symmetric transformation, [+] V = (T-iH)(T-Hil)-1 is an

**isometric**transformation (which is not necessarily everywhere defined). Show thefollowing: The operator T is closed if and only if V is closed. The operator I–V is ...

Page 1272

It may be proved that or and 3 are the same manifolds introduced in Definition 4.9

, and that an

= 0 = d . Also it is clear that a closed

It may be proved that or and 3 are the same manifolds introduced in Definition 4.9

, and that an

**isometric**operator V is unitary if and only if Q(V) = S) = }{(V), i.e., if d1= 0 = d . Also it is clear that a closed

**isometric**operator has a unitary extension ...Page 1372

... are p-continuous, and that the corresponding matrix of densities is {},(2)} = X o,

0) , lk 1 Consequently, if we put (Bf),(A) = X. Ibo (2)f(Z) for each n-tuple F = [fi, ..., f,

of Borel functions, B is an

... are p-continuous, and that the corresponding matrix of densities is {},(2)} = X o,

0) , lk 1 Consequently, if we put (Bf),(A) = X. Ibo (2)f(Z) for each n-tuple F = [fi, ..., f,

of Borel functions, B is an

**isometric**isomorphism of L2(A, (6,}) into L2(A, {p,3).### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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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