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 μ and a non - zero vector æ 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 μ and a non - zero vector æ 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 , Tx → x for every x in D ( T ) . By Theorem 2.9 ( b ) we have [ * ] ∞ o ( T ) ~ { 0 } 2Ŭ o ( T , ) 2 o ( T ) . n n = 1 Thus , if o ( T ) C [ 0 , ∞ ) , it follows from Theorem X.4.2 that T2 ≥ 0 . Hence ( Tx , x ) = lim ...
... Theorem 2.3 , Tx → x for every x in D ( T ) . By Theorem 2.9 ( b ) we have [ * ] ∞ o ( T ) ~ { 0 } 2Ŭ o ( T , ) 2 o ( T ) . n n = 1 Thus , if o ( T ) C [ 0 , ∞ ) , it follows from Theorem X.4.2 that T2 ≥ 0 . Hence ( Tx , x ) = lim ...
Page 1357
... follows from Theorem IV.8.1 that S ( SAV ) ( 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 S ( SAV ) ( 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 ) ...
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BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero