## Linear Operators: Spectral theory |

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Page 899

... functions on a(T) into the B* -algebra of bounded o-perators in Hilbert space

with the property that the

complex Borel functions converges at each point of a(T) to the

) ...

... functions on a(T) into the B* -algebra of bounded o-perators in Hilbert space

with the property that the

**functions f**(X) = X ... uniformly bounded sequence {/„} ofcomplex Borel functions converges at each point of a(T) to the

**function f**then fn(T) ...

Page 951

1 Lemma, (a) // / is X-measurable, then the

function. (b) For f, geL^R) the

and the convolution f*g of f and g, which is defined by the equation ...

1 Lemma, (a) // / is X-measurable, then the

**function f**(x—y) is a Xx X-measurablefunction. (b) For f, geL^R) the

**function f**(x—y)g{y) is integrable in y for almost all xand the convolution f*g of f and g, which is defined by the equation ...

Page 1207

Thus to prove the lemma it suffices to observe that no x ^ 0 is orthogonal to each

of the spaces £>„• Indeed, if x ^ 0 is orthogonal to the space ,£)a then, for a

bounded Borel

F(T)x, ...

Thus to prove the lemma it suffices to observe that no x ^ 0 is orthogonal to each

of the spaces £>„• Indeed, if x ^ 0 is orthogonal to the space ,£)a then, for a

bounded Borel

**function F**and a point y in £)0 we see, from Theorem 2.6(d), that (F(T)x, ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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Acad adjoint extension adjoint operator algebra Amer analytic B*-algebra Banach spaces Borel set boundary conditions boundary values bounded operator closed closure coefficients complex numbers constant 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 q Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma Proc prove real axis real numbers representation satisfies Section sequence singular solution spectral spectral theory square-integrable subspace Suppose symmetric operator theory topology transform unique unitary vanishes vector zero