Linear Operators: Spectral theory |
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Page 1290
... formally self adjoint provided only that the coefficients p¿ 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 p¿ 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 To ( t ) is symmetric . PROOF . Clearly To ( t ) CT1 ( t ) . Corollary 5 shows T1 ( t ) ≤ To ( t ) * . Q.E.D. τ that We recall ( cf. Definition XII.4.9 ) that if 7 is formally self adjoint , the ...
... formally self adjoint then the operator To ( t ) is symmetric . PROOF . Clearly To ( t ) CT1 ( t ) . Corollary 5 shows T1 ( t ) ≤ To ( t ) * . Q.E.D. τ that We recall ( cf. Definition XII.4.9 ) that if 7 is formally self adjoint , the ...
Page 1460
... formal differential operator . T T 34 THEOREM . Let τ be a formally self adjoint formal differential operator of order n on an interval I , and suppose that τ is bounded below . Let T1 be a regular or irregular differential operator of ...
... formal differential operator . T T 34 THEOREM . Let τ be a formally self adjoint formal differential operator of order n on an interval I , and suppose that τ is bounded below . Let T1 be a regular or irregular differential operator of ...
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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