## Linear Operators: Spectral operators |

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Page 1036

Q.E.D. 26 THEOREM. Let T be a Hilbert-Schmidt operator with non-zero

eigenvalues A1, A2, ... repeated according to multiplicities. Then the infinite

product co A. q'A(T) = TI ( - #) exi/A it:1 A.

analytic for A # 0.

Q.E.D. 26 THEOREM. Let T be a Hilbert-Schmidt operator with non-zero

eigenvalues A1, A2, ... repeated according to multiplicities. Then the infinite

product co A. q'A(T) = TI ( - #) exi/A it:1 A.

**converges**and defines a functionanalytic for A # 0.

Page 1436

Let {g,} be a bounded sequence of elements of Q(T) such that {Tg,}

Find a subsequence {g,} = {h} such that r”(h,)

h, → h, →X; or: (h)o, is in Q, and Th, – Th,. Thus (h.)

X.

Let {g,} be a bounded sequence of elements of Q(T) such that {Tg,}

**converges**.Find a subsequence {g,} = {h} such that r”(h,)

**converges**for each j, I < j < k. Thenh, → h, →X; or: (h)o, is in Q, and Th, – Th,. Thus (h.)

**converges**, so that {h,} = {h,+X.

Page 1664

The Fourier series of an element F in D, (C)

. It follows from the Definition 37 of the topology in D.(C) that it suffices to show

that (2+)-” X Fos e”q.(r)dr L C.

The Fourier series of an element F in D, (C)

**converges**unconditionally to F. Proof. It follows from the Definition 37 of the topology in D.(C) that it suffices to show

that (2+)-” X Fos e”q.(r)dr L C.

**converges**unconditionally to F(q) for each p in C.### What people are saying - Write a review

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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