## Linear Operators: Spectral theory |

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Page 1241

Nelson Dunford, Jacob T. Schwartz. then , letting 2n = Xn - Yn , we have lim.no %

n = 0 and limm , n - 12m - 2m / + = 0.

Izml + SM , m = 1 , 2 , .... Moreover , given ε > 0 there is an integer N such that if ...

Nelson Dunford, Jacob T. Schwartz. then , letting 2n = Xn - Yn , we have lim.no %

n = 0 and limm , n - 12m - 2m / + = 0.

**Consequently**there is a number Msuch thatIzml + SM , m = 1 , 2 , .... Moreover , given ε > 0 there is an integer N such that if ...

Page 1383

With boundary conditions A , the eigenvalues are

from the equation sin vă = 0 .

the numbers of the form ( na ) , n 2 1 ; in Case C , the numbers { ( n + 1 ) a } ?, n 2

...

With boundary conditions A , the eigenvalues are

**consequently**to be determinedfrom the equation sin vă = 0 .

**Consequently**, in Case A , the eigenvalues 1 arethe numbers of the form ( na ) , n 2 1 ; in Case C , the numbers { ( n + 1 ) a } ?, n 2

...

Page 1387

with the kernel sin Vās ( cos Vāt + i sin Vāt ) s < t , 32 > 0 vā t < s , In > , sin Vāt (

cos Văs + i sin Vās ) va sin Văs ( cos vă - i sin Vāt ) v V2 sin Vāt ( cos Vās i sin

Vās ) ...

**Consequently**, by Theorem 3.16 , the resolvent R ( 2 ; T ) is an integral operatorwith the kernel sin Vās ( cos Vāt + i sin Vāt ) s < t , 32 > 0 vā t < s , In > , sin Vāt (

cos Văs + i sin Vās ) va sin Văs ( cos vă - i sin Vāt ) v V2 sin Vāt ( cos Vās i sin

Vās ) ...

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