## Linear Operators: Spectral theory |

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Page 1010

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

a

is ...

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

**HilbertSchmidt****operators**may be defined as follows. ... A bounded linear operator T is said to bea

**Hilbert**-**Schmidt operator**in case the quantity |T| defined by the equation T-X, tois ...

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 that in ...

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 that in ...

Page 1132

If K is a

kernels representing K in the sense that (3) Kf1(s), fz(s), . . ..] — gi(s), g2(8), . . .]

where co ("1 (4) g,(s) = X s Ku(s, t)f,(t)dt, j=1 J0 the series converging

unconditionally in ...

If K is a

**Hilbert**-**Schmidt operator**in L2(A), there exists a unique set K,(s,t) ofkernels representing K in the sense that (3) Kf1(s), fz(s), . . ..] — gi(s), g2(8), . . .]

where co ("1 (4) g,(s) = X s Ku(s, t)f,(t)dt, j=1 J0 the series converging

unconditionally in ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Spectral Representation | 909 |

Copyright | |

17 other sections not shown

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adjoint extension adjoint operator algebra Amer analytic B-algebra Banach Borel set boundary conditions boundary values bounded operator closed closure Cº(I coefficients complete complex numbers 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 f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel Lemma Let f linear operator linearly independent mapping Math matrix measure Nauk SSSR N.S. neighborhood norm open set operators in Hilbert orthogonal orthonormal Plancherel's theorem positive Proc PRoof prove real numbers satisfies sequence singular ſº solution spectral spectral set spectral theory square-integrable subspace Suppose theory To(r topology transform unique unitary vanishes vector zero