## 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 1250

Finally we show that the decomposition T = PA of the theorem is

Lemma ... Since A is

of P(Air) = Tr. Further the extension of P by continuity from SR(A) to R(A) is

Finally we show that the decomposition T = PA of the theorem is

**unique**. ByLemma ... Since A is

**unique**, P is**uniquely**determined on SR(A) by the equationof P(Air) = Tr. Further the extension of P by continuity from SR(A) to R(A) is

**unique**.Page 1283

... denote the norm of an operator A in Euclidean n-space, and where K = maxies

A(t). Thus, if we use the norm |Y =s, Y()d in the B-space {L1(I)}", we have q)*| <

Kosk!, where K1 = K • maxler st–tol. Thus, equation (e') has the

cf.

... denote the norm of an operator A in Euclidean n-space, and where K = maxies

A(t). Thus, if we use the norm |Y =s, Y()d in the B-space {L1(I)}", we have q)*| <

Kosk!, where K1 = K • maxler st–tol. Thus, equation (e') has the

**unique**solution (cf.

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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