## Linear Operators: Spectral operators |

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Page 1083

Consequently, the series A(2)–26(A)I = XA, A" n=2 of the preceding

converges in the Hilbert-Schmidt norm. 44 Let (S, 2, u) be a positive measure

space. Then an operator A in the Hilbert space L2(S, 2, u) is of Hilbert-Schmidt

class if ...

Consequently, the series A(2)–26(A)I = XA, A" n=2 of the preceding

**exercise**converges in the Hilbert-Schmidt norm. 44 Let (S, 2, u) be a positive measure

space. Then an operator A in the Hilbert space L2(S, 2, u) is of Hilbert-Schmidt

class if ...

Page 1086

is the determinant of the (n+1)×(n+1) matrix whose elements are given by the

formulae au = 4 (s,t); 21, =4(s, sy-1), m + 1 2 j > 1: 2, 1 = 4(s;-1, t): m + 1 2 j > 1; 2,

- A(8, 1, 3,-1), n+1 = i, j > 1: determines a kernel satisfying (i) of

which ...

is the determinant of the (n+1)×(n+1) matrix whose elements are given by the

formulae au = 4 (s,t); 21, =4(s, sy-1), m + 1 2 j > 1: 2, 1 = 4(s;-1, t): m + 1 2 j > 1; 2,

- A(8, 1, 3,-1), n+1 = i, j > 1: determines a kernel satisfying (i) of

**Exercise**44,which ...

Page 1087

(Hint: For (d), use Weyl's inequality,

(Halberg) Let (S, X, u) be a g-finite measure space. Let T, be a 1-parameter family

of bounded operators defined in a subinterval I of the parameter interval i < p ...

(Hint: For (d), use Weyl's inequality,

**Exercise**30.) E. Miscellaneous Earercises 50(Halberg) Let (S, X, u) be a g-finite measure space. Let T, be a 1-parameter family

of bounded operators defined in a subinterval I of the parameter interval i < p ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

4 Exercises | 879 |

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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