## Linear Operators: Spectral theory |

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Page 1083

... an integral operator with a

then it is unique, and ... Let An(s, t) be the

in the sense of Exercise 44, the operator An of Exercise 42. Then the power

series ...

... an integral operator with a

**kernel**satisfying (i). If such a**kernel**A(', •) exists,then it is unique, and ... Let An(s, t) be the

**kernel**of Exercise 44 which represents,in the sense of Exercise 44, the operator An of Exercise 42. Then the power

series ...

Page 1130

Conversely, if Ktj is any family of

Hilbert-Schmidt operator in £)0 with norm given by (iv). ... by a unique

) e L2(A xA) in such a way that (1) ||JST||« = jAjA \K(a, b)\*v(daMdb) < oo and ...

Conversely, if Ktj is any family of

**kernels**satisfying (i), . . ., (iv), then (v) defines aHilbert-Schmidt operator in £)0 with norm given by (iv). ... by a unique

**kernel**K(', •) e L2(A xA) in such a way that (1) ||JST||« = jAjA \K(a, b)\*v(daMdb) < oo and ...

Page 1131

from Theorem III.2.20, Theorem 111.11.17, and Schwarz' inequality; thus the

integral on the right of (2) defines a bounded operator R. It is plain from the

definition of the

and by ...

from Theorem III.2.20, Theorem 111.11.17, and Schwarz' inequality; thus the

integral on the right of (2) defines a bounded operator R. It is plain from the

definition of the

**kernel**K that for each i and j. Thus, forming linear combinationsand by ...

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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