## Linear Operators: Spectral operators |

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Page 1010

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

a

Tr,”)} ...

resentations as an Le-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| = {XTr,”)} ...

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) Klf1(s), f(s), . . .] = [g(s), ga(s), . . .]

where (4) g,(s) – s 'K,', t)f(t)dt, j=1 J 0 the series converging unconditionally in the

...

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) Klf1(s), f(s), . . .] = [g(s), ga(s), . . .]

where (4) g,(s) – s 'K,', t)f(t)dt, j=1 J 0 the series converging unconditionally in the

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

Copyright | |

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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