Linear Operators: Spectral theory |
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Page 1041
... Theorem 6 , S is compact , it follows from Theorem VII.4.5 that there exists a non- zero complex number u and a non - zero vector a in sp ( T ) such that Sx = ux . Thus , by Theorem VII.4.5 again , E ( μ ; T * ) ( sp ( T ) ' ) # { 0 } . It ...
... Theorem 6 , S is compact , it follows from Theorem VII.4.5 that there exists a non- zero complex number u and a non - zero vector a in sp ( T ) such that Sx = ux . Thus , by Theorem VII.4.5 again , E ( μ ; T * ) ( sp ( T ) ' ) # { 0 } . It ...
Page 1247
... Theorem 2.3 , T , xx for every x in D ( T ) . By Theorem 2.9 ( b ) we have [ * ] ∞ o ( T ) ~ { 0 } 2 Ů o ( T , „ ) 2 0 ( T ) . n = 1 Thus , if o ( T ) C [ 0 , ∞ ) , it follows from Theorem X.4.2 that T≥ 0 . Hence ( Tx , x ) = lim ...
... Theorem 2.3 , T , xx for every x in D ( T ) . By Theorem 2.9 ( b ) we have [ * ] ∞ o ( T ) ~ { 0 } 2 Ů o ( T , „ ) 2 0 ( T ) . n = 1 Thus , if o ( T ) C [ 0 , ∞ ) , it follows from Theorem X.4.2 that T≥ 0 . Hence ( Tx , x ) = lim ...
Page 1357
... follows from Theorem IV.8.1 that fr | ( SVƒ ) ( t ) | 2 dt < ∞ , and that [ ** ] holds for all g in L2 ( I ) . Thus ... theorem . To prove ( ii ) , we argue as follows ( compare the proof of Corollary 3 ) . From the boundedness of G ( T ) ...
... follows from Theorem IV.8.1 that fr | ( SVƒ ) ( t ) | 2 dt < ∞ , and that [ ** ] holds for all g in L2 ( I ) . Thus ... theorem . To prove ( ii ) , we argue as follows ( compare the proof of Corollary 3 ) . From the boundedness of G ( T ) ...
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