## Linear Operators: Spectral theory |

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Page 868

S is isometrically isomorphic to the field of compler numbers if and only if 3 is

marimal. PROOF. If & is not ... It is seen from Theorem 1 that for each JJ e -o and

each as in 3: there is a

S is isometrically isomorphic to the field of compler numbers if and only if 3 is

marimal. PROOF. If & is not ... It is seen from Theorem 1 that for each JJ e -o and

each as in 3: there is a

**complex number**a (JR) such that a H-S) = w(Jo)e-i-J'.Page 902

Q.E.D. 10 CoRoll.ARY. Let fe EB(S, 2') where E is the spectral measure of

Corollary 9, and let E, be the resolution of the identity for the operator T(f). Then,

for every Borel set 6 of

essentially ...

Q.E.D. 10 CoRoll.ARY. Let fe EB(S, 2') where E is the spectral measure of

Corollary 9, and let E, be the resolution of the identity for the operator T(f). Then,

for every Borel set 6 of

**complex numbers**, E,(0) = E(f-(6)). PROOF. Since f isessentially ...

Page 1251

It!"u(dt) < 00 and - mn – so t"u(dt), n = 0, 1, 2, ..., —co is that n X m, 1, 2, 3, 20 i, j=0

for every finite set oo, ..., x, of

condition is necessary. For if {m,} has such a representation, and 20, ..., x, is any ...

It!"u(dt) < 00 and - mn – so t"u(dt), n = 0, 1, 2, ..., —co is that n X m, 1, 2, 3, 20 i, j=0

for every finite set oo, ..., x, of

**complex numbers**. PRoof. We observe first that thecondition is necessary. For if {m,} has such a representation, and 20, ..., x, is any ...

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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