## Linear Operators: Spectral operators |

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Page 1050

We begin our formal development by considering a Lebesgue measurable

of “singularities” at which it is not Lebesgue integrable, and defining a certain ...

We begin our formal development by considering a Lebesgue measurable

**function**f**defined**on Euclidean n-space E", supposing that f has a finite numberof “singularities” at which it is not Lebesgue integrable, and defining a certain ...

Page 1074

is the Fourier transform of a function in Lp(–oo, + oy whenever F is the Fourier

transform of a function in Lp(– o, + Co.), the Fourier transforms being defined as

in Exercise 6. 10 Let à be a

...

is the Fourier transform of a function in Lp(–oo, + oy whenever F is the Fourier

transform of a function in Lp(– o, + Co.), the Fourier transforms being defined as

in Exercise 6. 10 Let à be a

**function defined**on (– 60, -i- oc) which is of finite total...

Page 1645

vo. go airo to by !o: 2 (Aof)(a, y) = (r, y). ôa dy This is an operator densely

in L2(E*), but not a closed operator. If we let A be its closure, we find that D(A)

contains nondifferentiable

vo. go airo to by !o: 2 (Aof)(a, y) = (r, y). ôa dy This is an operator densely

**defined**in L2(E*), but not a closed operator. If we let A be its closure, we find that D(A)

contains nondifferentiable

**functions**. Which non-differentiable**functions**?### What people are saying - Write a review

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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