## Linear Operators: Spectral operators |

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Page 890

One class of

has already been defined is the class ... determined by T provided that we require

the

One class of

**scalar**functions f, other than polynomials, for which the operator f(T)has already been defined is the class ... determined by T provided that we require

the

**scalar**function g(z) = 2 and the operator T be corresponding elements.Page 1178

Then of maps

from Plancherel's theorem that 3% is a bounded mapping of the space L2 of

Then of maps

**scalar**-valued functions into functions with values in le. It is plainfrom Plancherel's theorem that 3% is a bounded mapping of the space L2 of

**scalar**-valued functions into the space L2(l,) of square-integrable vector-valued ...Page 1782

If, however, each of the spaces 3:1, ..., &, are Hilbert spaces then it will always be

understood, sometimes without explicit mention, that 3 is the uniquely deter:

mined Hilbert space with

9,), ...

If, however, each of the spaces 3:1, ..., &, are Hilbert spaces then it will always be

understood, sometimes without explicit mention, that 3 is the uniquely deter:

mined Hilbert space with

**scalar**product (iv) (sa'i, • * * * *...], [91, • * * * !/..]) F. X (a,9,), ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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### Common terms and phrases

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