Linear Operators: Spectral theory |
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Page 1019
The inequality is obvious for n = 1 and may be easily checked for n = 2. We shall suppose the inequality to be known for n - 1 , and proceed by induction . If ( aij ) is an nxn matrix , let U ; = { anj , Anja ani ] , į 1 , ...
The inequality is obvious for n = 1 and may be easily checked for n = 2. We shall suppose the inequality to be known for n - 1 , and proceed by induction . If ( aij ) is an nxn matrix , let U ; = { anj , Anja ani ] , į 1 , ...
Page 1094
To prove ( b ) , we put T = T1 + T , and note that , by Corollary 3 , M2n + 1 ( T1 + T2 ) = Mn + 2 ( 71 ) + Mm + 1 ( T2 ) M2n + 2 ( T , + T , ) Min + 1 ( 77 ) + Mn + 2 ( T2 ) . First let p 21. Then by Minkowski's inequality , 2 1 2 1 ...
To prove ( b ) , we put T = T1 + T , and note that , by Corollary 3 , M2n + 1 ( T1 + T2 ) = Mn + 2 ( 71 ) + Mm + 1 ( T2 ) M2n + 2 ( T , + T , ) Min + 1 ( 77 ) + Mn + 2 ( T2 ) . First let p 21. Then by Minkowski's inequality , 2 1 2 1 ...
Page 1105
We now pause to sharpen another of the inequalities of Lemma 9 . ... the continuity of the norm function which follows from the triangle inequality of Lemma 14 ( d ) , and by Lemma 11 , it follows that we may without loss of generality ...
We now pause to sharpen another of the inequalities of Lemma 9 . ... the continuity of the norm function which follows from the triangle inequality of Lemma 14 ( d ) , and by Lemma 11 , it follows that we may without loss of generality ...
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Contents
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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