## Linear Operators: General theory |

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Page 565

25 Let Y ( t ) be a

Show that the set of all non - singular matrix

Y ( t ) C where C is any nxn constant , nonsingular matrix . 26 Let A ( t ) have

period p ...

25 Let Y ( t ) be a

**solution**matrix of dY / dt = A ( t ) Y which is non - singular .Show that the set of all non - singular matrix

**solutions**are precisely the matricesY ( t ) C where C is any nxn constant , nonsingular matrix . 26 Let A ( t ) have

period p ...

Page 610

have unique

equations ( H ) ( 21 — T ) x = 0 , ( H * ) ( 21 – T * ) * * = 0 have only the zero

have unique

**solutions**for any y in X or y * in X * if and only if the homogeneousequations ( H ) ( 21 — T ) x = 0 , ( H * ) ( 21 – T * ) * * = 0 have only the zero

**solutions**. Furthermore if one of the homogeneous equations has a non - zero**solution**...Page 777

. S . ) 76 , 21 - 24 ( 1951 ) . ( Russian ) Math . Rev . 12 , 613 ( 1951 ) . On certain

problems on the maximum and minimum of characteristic values and on the ...

**Solution**of the inverse Sturm - Liouville problem . Doklady Akad . Nauk SSSR ( N. S . ) 76 , 21 - 24 ( 1951 ) . ( Russian ) Math . Rev . 12 , 613 ( 1951 ) . On certain

problems on the maximum and minimum of characteristic values and on the ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero