## Linear Operators: Spectral operators |

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Page 1126

of the closed set C; we shall

Since each projection in the spectral resolution of T and hence each continuous

function of T is a strong limit of linear combinations of the projections E, ...

of the closed set C; we shall

**denote**this subspace of L2[0, 1] by the symbol I,(C).Since each projection in the spectral resolution of T and hence each continuous

function of T is a strong limit of linear combinations of the projections E, ...

Page 1579

... where t is a formally symmetric formal differential operator of order n defined on

I, and where r is a positive function which is infinitely often differentiable on I. (a)

Let L2(I, r)

... where t is a formally symmetric formal differential operator of order n defined on

I, and where r is a positive function which is infinitely often differentiable on I. (a)

Let L2(I, r)

**denote**the set of measurable functions defined on I for which fo = I, ...Page 1661

Then rB' will

p e C. (I). (ii) The symbol F will

equation F(q) = F(j), q e C. (I). (iii) If Io is an open subset of I then the restriction

FIo will ...

Then rB' will

**denote**the element of D,(I) defined by the equation (1 F)(q) = F(t+p),p e C. (I). (ii) The symbol F will

**denote**the element of D,(I) defined by theequation F(q) = F(j), q e C. (I). (iii) If Io is an open subset of I then the restriction

FIo will ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

52 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### 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