Linear Operators: Spectral theory |
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Page 1247
... unique positive " square root " . 3 LEMMA . If T is a positive self adjoint transformation , there is a unique positive self adjoint transformation A such that A2 T. = = PROOF . By Lemma 2 , o ( T ) ≤ [ 0 , ∞ ) and , by Theorem 2.6 ...
... unique positive " square root " . 3 LEMMA . If T is a positive self adjoint transformation , there is a unique positive self adjoint transformation A such that A2 T. = = PROOF . By Lemma 2 , o ( T ) ≤ [ 0 , ∞ ) and , by Theorem 2.6 ...
Page 1250
... unique , P is uniquely determined on R ( A ) by the equation of P ( Ax ) Tx . Further the extension of P by continuity from R ( 4 ) to R ( A ) is unique . Since P is zero on R ( A ) 1 it follows that P is uniquely determined by ...
... unique , P is uniquely determined on R ( A ) by the equation of P ( Ax ) Tx . Further the extension of P by continuity from R ( 4 ) to R ( A ) is unique . Since P is zero on R ( A ) 1 it follows that P is uniquely determined by ...
Page 1378
... unique , and 0 , if i k or j > k . Pij = i , j Pij = 1 , k ; Pii - Ok PROOF . Suppose that σ1 , ... , σ is a determining set for T. Then it is evident from Theorem 23 that if we define { p1 ; } , i , j = 1 ,. .. . , n , by [ * ] Pij ( e ) ...
... unique , and 0 , if i k or j > k . Pij = i , j Pij = 1 , k ; Pii - Ok PROOF . Suppose that σ1 , ... , σ is a determining set for T. Then it is evident from Theorem 23 that if we define { p1 ; } , i , j = 1 ,. .. . , n , by [ * ] Pij ( e ) ...
Contents
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