## Linear Operators: Spectral theory |

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Page 1272

IX], Riesz and Sz.-Nagy [1; Sec. 123] and Ahiezer and Glazman [1; Secs. 78–80].

Marimal symmetric operators. If T is a symmetric operator with dense domain,

then it has proper symmetric extensions provided both of its deficiency

are ...

IX], Riesz and Sz.-Nagy [1; Sec. 123] and Ahiezer and Glazman [1; Secs. 78–80].

Marimal symmetric operators. If T is a symmetric operator with dense domain,

then it has proper symmetric extensions provided both of its deficiency

**indices**are ...

Page 1398

Q.E.D. REMARK. The assumption that A does not belong to the essential

spectrum in Corollary 8 is necessary. For example, if r = –(d/dt)* on the interval [0,

oo), both deficiency

A > 0 ...

Q.E.D. REMARK. The assumption that A does not belong to the essential

spectrum in Corollary 8 is necessary. For example, if r = –(d/dt)* on the interval [0,

oo), both deficiency

**indices**of r may readily be seen to be 1. On the other hand, ifA > 0 ...

Page 1454

If T is a closed symmetric operator in Hilbert space, and T is bounded below, then

(a) the essential spectrum of T is a subset of the real aris which is bounded below

; (b) the deficiency

If T is a closed symmetric operator in Hilbert space, and T is bounded below, then

(a) the essential spectrum of T is a subset of the real aris which is bounded below

; (b) the deficiency

**indices**of T are equal. PRoof. To prove (a), note that if T is ...### What people are saying - Write a review

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f 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 singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero