## Linear Operators: Spectral theory |

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Page 1036

Then the infinite product

each fixed X 0, <px(T) is a continuous complex valued function on the B-space of

all Hilbert-Schmidt operators. Proof. First note that if f is a complex number with |f

...

Then the infinite product

**converges**and defines a function analytic for X ^ 0. Foreach fixed X 0, <px(T) is a continuous complex valued function on the B-space of

all Hilbert-Schmidt operators. Proof. First note that if f is a complex number with |f

...

Page 1436

Let {gn} be a bounded sequence of elements of such that {Tgn}

a subsequence {gn } = {h(} such that xf(h()

hi—'^_1x*(hi)q>j is in 5), and TK( = Thf. Thus {ht}

Let {gn} be a bounded sequence of elements of such that {Tgn}

**converges**. Finda subsequence {gn } = {h(} such that xf(h()

**converges**for each /,1 ^j^k. Then h( =hi—'^_1x*(hi)q>j is in 5), and TK( = Thf. Thus {ht}

**converges**, so that {h-} ...Page 1664

The Fourier series of an element F in Dn(C)

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

that (2rc)~n2F£ I eiL x<p(x)dx L JC

...

The Fourier series of an element F in Dn(C)

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

that (2rc)~n2F£ I eiL x<p(x)dx L JC

**converges**unconditionally to ^(9?) for each cp...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 860 |

Commutative BAlgebras | 874 |

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Acad adjoint extension adjoint operator algebra Amer analytic B-algebra 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 g Haar measure Hence Hilbert space Hilbert-Schmidt operator hypothesis identity inequality integral interval kernel Lemma linear operator linearly independent mapping Math matrix measure Nauk SSSR N. S. neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive Proc prove real axis real numbers representation satisfies second order Section sequence singular solution spectral set spectral theory square-integrable subspace Suppose symmetric operator topology transform unique unitary vanishes vector zero