Linear Operators, Part 2 |
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Page 1290
... formally self adjoint provided only that the coefficients pi are real . In the same way , the formal differential operator ( i / 2 ) ( d / dt ) " { p ( t ) ( d / dt ) + ( d / dt ) p ( t ) } ( d / dt ) " is formally self adjoint provided ...
... formally self adjoint provided only that the coefficients pi are real . In the same way , the formal differential operator ( i / 2 ) ( d / dt ) " { p ( t ) ( d / dt ) + ( d / dt ) p ( t ) } ( d / dt ) " is formally self adjoint provided ...
Page 1295
... formally self adjoint then the operator Tô ( τ ) is symmetric . Corollary 5 shows PROOF . Clearly T1 ( T ) To ( T ) * . Q.E.D. To ( t ) CT1 ( t ) . that We recall ( cf. Definition XII.4.9 ) that if 7 is formally self adjoint , the ...
... formally self adjoint then the operator Tô ( τ ) is symmetric . Corollary 5 shows PROOF . Clearly T1 ( T ) To ( T ) * . Q.E.D. To ( t ) CT1 ( t ) . that We recall ( cf. Definition XII.4.9 ) that if 7 is formally self adjoint , the ...
Page 1503
... formally self adjoint formal differential operator to the normal form in d dt n This leads to the normal form + n - 2 di Σa , ( t ) ( a ) 2 ' dt j = 0 d dt for the operator of order 1 , so that every formally self adjoint formal ...
... formally self adjoint formal differential operator to the normal form in d dt n This leads to the normal form + n - 2 di Σa , ( t ) ( a ) 2 ' dt j = 0 d dt for the operator of order 1 , so that every formally self adjoint formal ...
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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