## Linear Operators: Spectral theory |

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Page 1305

We conclude this section by

operators. The simplest example of a formally self ... We shall

choices for the interval I. Case 1: I = [0, 1]. Here clearly d, = d = 1, and a complete

set of ...

We conclude this section by

**considering**some simple examples of differentialoperators. The simplest example of a formally self ... We shall

**consider**threechoices for the interval I. Case 1: I = [0, 1]. Here clearly d, = d = 1, and a complete

set of ...

Page 1408

, b), with norm f = lub,<!-, f(t). In this space

transformation: b . (Mi)() = s.sin (s—t)q(s)f(s)as. Since (Mf)(t) s |f| s: |q(s)|ds, we

have |M| ...

**Consider**the B-space C Bar, b) of all bounded continuous functions defined on [a, b), with norm f = lub,<!-, f(t). In this space

**consider**the following lineartransformation: b . (Mi)() = s.sin (s—t)q(s)f(s)as. Since (Mf)(t) s |f| s: |q(s)|ds, we

have |M| ...

Page 1682

Now

XI.3.1 shows that equation (iv) defines a continuous map a -> c from L1(E") into

L1(E") as well as one from Ls (E") into L. (E"). It follows from the Riesz convexity ...

Now

**consider**the convolution (iv) c(u)= s.a(r)/(r-u)du. If b is in L1(E") then LemmaXI.3.1 shows that equation (iv) defines a continuous map a -> c from L1(E") into

L1(E") as well as one from Ls (E") into L. (E"). It follows from the Riesz convexity ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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### 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

additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero