Linear Operators: Spectral theory |
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Page 1295
... To ( t ) is symmetric . PROOF . Clearly To ( t ) CT ( T ) . Corollary T1 ( t ) . T1 ( t ) ≤ To ( t ) * . Q.E.D. τ 5 shows that We recall ( cf. Definition XII.4.9 ) that if 7 is formally self adjoint , the positive and negative ...
... To ( t ) is symmetric . PROOF . Clearly To ( t ) CT ( T ) . Corollary T1 ( t ) . T1 ( t ) ≤ To ( t ) * . Q.E.D. τ 5 shows that We recall ( cf. Definition XII.4.9 ) that if 7 is formally self adjoint , the positive and negative ...
Page 1303
... ( To ( T ) ) we have [ * ] 0 = ( f , g ) * ( f , g ) + ( To ( t ) f , T1 ( t ) g ) . It follows , as in Corollary 22 , that g is a solution of the equation 0 , and is therefore infinitely differentiable . Let v = −īg ; T * Tg + g then ...
... ( To ( T ) ) we have [ * ] 0 = ( f , g ) * ( f , g ) + ( To ( t ) f , T1 ( t ) g ) . It follows , as in Corollary 22 , that g is a solution of the equation 0 , and is therefore infinitely differentiable . Let v = −īg ; T * Tg + g then ...
Page 1437
... To ( t ) ≤ T1 ( t ) , it follows immediately from the preceding lemma that 2 € σ , ( T1 ( t ) ) , so that by ... ( To ( t ) ) → D. Moreover , if we have fe D , then ( f , g ) + ( T1 ( t ) f , To ( t ) g ) = 0 for g in D ( To ( 7 ) ...
... To ( t ) ≤ T1 ( t ) , it follows immediately from the preceding lemma that 2 € σ , ( T1 ( t ) ) , so that by ... ( To ( t ) ) → D. Moreover , if we have fe D , then ( f , g ) + ( T1 ( t ) f , To ( t ) g ) = 0 for g in D ( To ( 7 ) ...
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BAlgebras | 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 complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero