## Linear Operators: Spectral theory |

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Page 1187

The inverse of a

only if its domain is

which maps ( x , y ) into [ y , x ] then I ( T - 1 ) = A , ( T ) which shows that T is ...

The inverse of a

**closed**operator is**closed**. A bounded operator is**closed**if andonly if its domain is

**closed**. PROOF . If A , is the isometric automorphism in H O Hwhich maps ( x , y ) into [ y , x ] then I ( T - 1 ) = A , ( T ) which shows that T is ...

Page 1393

We begin by defining a certain type of “ pectrum ” for the formal differential

operator t . 1 DEFINITION . Let T be a

set of complex numbers 2 such that the range of 21 – T is not

...

We begin by defining a certain type of “ pectrum ” for the formal differential

operator t . 1 DEFINITION . Let T be a

**closed**operator in Hilbert space . Then theset of complex numbers 2 such that the range of 21 – T is not

**closed**is called the...

Page 1436

2 , every finite dimensional subspace of a B - space is

HahnBanach theorem ( II . 3 . 13 ) there exists a set x * , . . . , xe of continuous

linear functionals on the B - space such that x * ( Q ; ) = 0 for 0 Si # i sk , x * ( qi ) =

1 .

2 , every finite dimensional subspace of a B - space is

**closed**. Thus , by theHahnBanach theorem ( II . 3 . 13 ) there exists a set x * , . . . , xe of continuous

linear functionals on the B - space such that x * ( Q ; ) = 0 for 0 Si # i sk , x * ( qi ) =

1 .

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### Contents

IX | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Compact Groups | 945 |

Copyright | |

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additive adjoint operator algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently consider constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero