## Linear Operators: Spectral operators |

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Page 1247

Hence, by Theorem X.4.2., o(T,) C [0, co). Thus [+] shows that g(T) C [0, oo).

Q.E.D. The next lemma shows that a positive self adjoint transformation has a

transformation, ...

Hence, by Theorem X.4.2., o(T,) C [0, co). Thus [+] shows that g(T) C [0, oo).

Q.E.D. The next lemma shows that a positive self adjoint transformation has a

**unique**positive “square root”. 8 LEMMA. If T is a positive self adjointtransformation, ...

Page 1283

Thus, equation (e') has the

be closed. It still follows from the proof given above that if J is any closed

subinterval of I containing the point to, there exists a

such that (a) ...

Thus, equation (e') has the

**unique**solution (cf. Lemma VII.3.4) co F ... known tobe closed. It still follows from the proof given above that if J is any closed

subinterval of I containing the point to, there exists a

**unique**function f, e A"(J)such that (a) ...

Page 1383

Again we are in the situation of Section 4, the interval being finite, the spectrum

being discrete, and the set of eigenfunctions being complete. With boundary

conditions A and C, the

condition ...

Again we are in the situation of Section 4, the interval being finite, the spectrum

being discrete, and the set of eigenfunctions being complete. With boundary

conditions A and C, the

**unique**solution of tao = Ao satisfying the boundarycondition ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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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