Linear Operators, Part 2 |
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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 ) it follows that P is uniquely determined by T. Q.E.D. ...
... 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 ) it follows that P is uniquely determined by T. Q.E.D. ...
Page 1283
... unique solution ( cf. Lemma VII.3.4 ) ∞ F = ( 1 + 0 ) 1H = Σ ( -1 ) ΦΗ . j = 0 Since all the terms in equation ( e ) but the first are absolutely contin- uous , it follows that F is absolutely continuous . Thus Theorem 1 is proved for ...
... unique solution ( cf. Lemma VII.3.4 ) ∞ F = ( 1 + 0 ) 1H = Σ ( -1 ) ΦΗ . j = 0 Since all the terms in equation ( e ) but the first are absolutely contin- uous , it follows that F is absolutely continuous . Thus Theorem 1 is proved for ...
Page 1378
... unique , and Pis Pis , i , j = 1 , ... , k ; p1 = 0 , if i > k or j > k . = Pii PROOF . Suppose that σ1 , . . . , σ is a determining set for T. Then it is evident from Theorem 23 that if we define { P¿¡ } , i , j = 1 , ... , N , by ...
... unique , and Pis Pis , i , j = 1 , ... , k ; p1 = 0 , if i > k or j > k . = Pii PROOF . Suppose that σ1 , . . . , σ is a determining set for T. Then it is evident from Theorem 23 that if we define { P¿¡ } , i , j = 1 , ... , N , by ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 domain 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(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero