Linear Operators: Spectral theory |
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Page 1260
17 ( Schmidt ) The non - zero eigenvalues of T * T are the same as the non - zero eigenvalues of TT * , even as to multiplicity ( the positive square roots of these eigenvalues are sometimes called the characteristic numbers of T ) .
17 ( Schmidt ) The non - zero eigenvalues of T * T are the same as the non - zero eigenvalues of TT * , even as to multiplicity ( the positive square roots of these eigenvalues are sometimes called the characteristic numbers of T ) .
Page 1464
Then ( a ) if lim supe - t - g ( t ) < - ( 1/4 ) , every solution of tf = 0 has 0 an infinite number of zeros on [ a , 0 ) ; ( b ) if lim infotag ( t ) > - ( 1/4 ) , no solution , not identically zero , of the 0 has more than a finite ...
Then ( a ) if lim supe - t - g ( t ) < - ( 1/4 ) , every solution of tf = 0 has 0 an infinite number of zeros on [ a , 0 ) ; ( b ) if lim infotag ( t ) > - ( 1/4 ) , no solution , not identically zero , of the 0 has more than a finite ...
Page 1727
By ( 1 ) and by the definitions ( 2 ) , ( 3 ) , and ( 5 ) of Si , it follows that L , > L L ' leht ( 7 ) ( S29 ) ( x ) = 0,9 EC ( E ” ) , -k Smin ( L ) Smax ( L ) Sk , if one of x1 , ... , X , is zero . Suppose that we let I , denote ...
By ( 1 ) and by the definitions ( 2 ) , ( 3 ) , and ( 5 ) of Si , it follows that L , > L L ' leht ( 7 ) ( S29 ) ( x ) = 0,9 EC ( E ” ) , -k Smin ( L ) Smax ( L ) Sk , if one of x1 , ... , X , is zero . Suppose that we let I , denote ...
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Contents
8 | 876 |
859 | 885 |
extensive presentation of applications of the spectral theorem | 911 |
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additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero