## Linear Operators: Spectral theory |

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Page 1010

resentations as an L2-space and in this setting the class of

orthonormal set in the Hilbert space §. A bounded linear operator T is said to be a

...

resentations as an L2-space and in this setting the class of

**Hilbert**-**Schmidt****operators**may be defined as follows. 1 Definition. Let {xx, a. e A} be a completeorthonormal set in the Hilbert space §. A bounded linear operator T is said to be a

...

Page 1013

The operator T is compact (cf. Exercise X.8.5), but it is not in HS. It has been

noted in the preceding discussion that the class of

forms a Banach algebra (without identity) under the norm || • |. It may readily be

shown ...

The operator T is compact (cf. Exercise X.8.5), but it is not in HS. It has been

noted in the preceding discussion that the class of

**Hilbert**-**Schmidt operators**forms a Banach algebra (without identity) under the norm || • |. It may readily be

shown ...

Page 1132

If K is a

kernels representing K in the sense that (3) ff[M*),/i(»)....] = [&(»), &(*),...] where (4)

«,(«) = I *„(*, <)/,(<)*, j=i Jo the series converging unconditionally in the topology

of ...

If K is a

**Hilbert**-**Schmidt operator**in L2{A), there exists a unique set Ku(s, t) ofkernels representing K in the sense that (3) ff[M*),/i(»)....] = [&(»), &(*),...] where (4)

«,(«) = I *„(*, <)/,(<)*, j=i Jo the series converging unconditionally in the topology

of ...

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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