## Linear Operators: Spectral theory |

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Page 1256

By Theorem II.1.18, this holds not only for a e 3, but also for a e S). Thus U(t) is a

strongly continuous group of unitary operators. It now

that there exists an unbounded (or possibly bounded) self adjoint operator T such

...

By Theorem II.1.18, this holds not only for a e 3, but also for a e S). Thus U(t) is a

strongly continuous group of unitary operators. It now

**follows from Theorem**6.1that there exists an unbounded (or possibly bounded) self adjoint operator T such

...

Page 1358

i, j=1 J e It

equation [*] holds for all f in L2(I). Q.E.D. 15 CoRoll ARY. Let T, A, and {p,} be

defined as in Theorem 14. The complement of q(T) in A is the largest open subset

eo of ...

i, j=1 J e It

**follows from Theorem**IV.8.l. that [s, K.G. . )*ds]'s M(j), tes, and thatequation [*] holds for all f in L2(I). Q.E.D. 15 CoRoll ARY. Let T, A, and {p,} be

defined as in Theorem 14. The complement of q(T) in A is the largest open subset

eo of ...

Page 1379

{6,6} is the matrix measure of

determined for each e C N. Since A is the union of a sequence of neighborhoods

of the same type as N, the uniqueness of {6,3

{6,6} is the matrix measure of

**Theorem**23, the values 6,(e) are uniquelydetermined for each e C N. Since A is the union of a sequence of neighborhoods

of the same type as N, the uniqueness of {6,3

**follows**immediately. Q.E.D. 27**THEOREM**.### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers continuous function converges Corollary deficiency indices Definition denote dense differential equations Doklady Akad domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero