## Linear Operators: Spectral operators |

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Page 898

If we put E(0) = 0 when Ó n g(T) is void, then

Theorem 1 and

spectral measure associated, in

If we put E(0) = 0 when Ó n g(T) is void, then

**Corollary**4 follows immediately fromTheorem 1 and

**Corollary**IX.3.15. Q.E.D. 5 DEFINITION. The uniquely definedspectral measure associated, in

**Corollary**4, with the normal operator T is called ...Page 1301

If the assertion of the

value at a which is independent of the set A0, ..., A, 1, and hence has at least n+1

independent boundary values at a. But this is impossible by

If the assertion of the

**corollary**were false, it would follow that t has a boundaryvalue at a which is independent of the set A0, ..., A, 1, and hence has at least n+1

independent boundary values at a. But this is impossible by

**Corollary**22.Page 1459

Thus the present

31 CoRoll.ARY. Suppose in addition to the hypotheses of Theorem 8 that the

coefficients p, are real. Then t is finite below any finite A. PRoof. We use the

notations ...

Thus the present

**corollary**follows from**Corollary**7 and Definition 25(b). Q.E.D.31 CoRoll.ARY. Suppose in addition to the hypotheses of Theorem 8 that the

coefficients p, are real. Then t is finite below any finite A. PRoof. We use the

notations ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

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adjoint extension adjoint operator algebra analytic B-algebra B-space Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality infinity integral interval kernel Lemma Let f Let Q linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real numbers satisfies sequence singular ſº solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory To(r To(t topology transform uniformly unique unitary vanishes vector zero