Linear Operators: Spectral theory |
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Page 1100
... prove ( b ) in general , we have only to prove ( b ) in the special case in which A and B have finite - dimensional ranges . But then the inequality in ( b ) is plainly a special case of ( a ) , while the identity ( 3 ) follows readily ...
... prove ( b ) in general , we have only to prove ( b ) in the special case in which A and B have finite - dimensional ranges . But then the inequality in ( b ) is plainly a special case of ( a ) , while the identity ( 3 ) follows readily ...
Page 1198
... prove ( d ) let x , y be in D ( J ( T ) ) = D ( ƒ ( T ) ) . Then ∞ ( Ƒ ( T ) x , y ) = [ _∞ % Ƒ ( ^ ) ( E ( dλ ) x , y ) = [ ~ ƒ ( 2 ) ( E ( dλ ) y , x ) = ( x , f ( T ) y ) . Thus ƒ ( T ) ≤ ƒ ( T ) * and to prove ( d ) it ...
... prove ( d ) let x , y be in D ( J ( T ) ) = D ( ƒ ( T ) ) . Then ∞ ( Ƒ ( T ) x , y ) = [ _∞ % Ƒ ( ^ ) ( E ( dλ ) x , y ) = [ ~ ƒ ( 2 ) ( E ( dλ ) y , x ) = ( x , f ( T ) y ) . Thus ƒ ( T ) ≤ ƒ ( T ) * and to prove ( d ) it ...
Page 1554
... Prove that the equation g ' ( t ) —q2 ( t ) g ( t ) = 0 has a square - integrable solution . ( Hint : Write g ( t ) = h ( t ) f ( t ) and obtain , by variation of para- meters , an equation of type ( A ) as in G7 for h . Infer that g ...
... Prove that the equation g ' ( t ) —q2 ( t ) g ( t ) = 0 has a square - integrable solution . ( Hint : Write g ( t ) = h ( t ) f ( t ) and obtain , by variation of para- meters , an equation of type ( A ) as in G7 for h . Infer that g ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero