Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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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 ( A ) to R ( 4 ) 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 ( A ) to R ( 4 ) 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 = ( I + Ø ) −1 H - ∞ Σ ( -1 ) ΦΗ . ( 1 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 ...
... unique solution ( cf. Lemma VII.3.4 ) F = ( I + Ø ) −1 H - ∞ Σ ( -1 ) ΦΗ . ( 1 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 ...
Page 1378
... unique , and Pij , i , j = 1 , ... , k ; p1 = 0 , if i > k or j > k . Pij = 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 [ * ] Pis ...
... unique , and Pij , i , j = 1 , ... , k ; p1 = 0 , if i > k or j > k . Pij = 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 [ * ] Pis ...
Contents
IX | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients compact subset complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood norm open set open subset orthonormal partial differential operator Plancherel's theorem positive PROOF prove real axis real numbers satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose T₁ T₁(t theory To(t topology unique unitary vanishes vector zero