## Linear Operators: Spectral theory |

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Page 888

where a, 6 are arbitrary

used the notations A A B and A v B for the intersection and union of two

commuting projections A and B. We recall that these operators are defined by the

...

where a, 6 are arbitrary

**spectral sets**and where q is the void set. Here we haveused the notations A A B and A v B for the intersection and union of two

commuting projections A and B. We recall that these operators are defined by the

...

Page 933

Halmos [9] also considers the relation of the spectra. The

Neumann. If T is a bounded linear operator in a Hilbert space, then von

Neumann [3] defines a closed set S of the complex sphere to be a

T if f(T) ...

Halmos [9] also considers the relation of the spectra. The

**spectral sets**of vonNeumann. If T is a bounded linear operator in a Hilbert space, then von

Neumann [3] defines a closed set S of the complex sphere to be a

**spectral set**ofT if f(T) ...

Page 993

If the bounded measurable function op has its

point m then, for some compler number 2, p(r) = xsa, m] for almost all a in R.

PRoof. In view of Lemma 11(d) it suffices to prove the theorem in the case m = 0.

If the bounded measurable function op has its

**spectral set**consisting of the singlepoint m then, for some compler number 2, p(r) = xsa, m] for almost all a in R.

PRoof. In view of Lemma 11(d) it suffices to prove the theorem in the case m = 0.

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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