Linear Operators, Part 2 |
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Page 1469
... preceding lemma . Q.E.D. Next we turn to the more detailed analysis of case ( b ) of the preceding theorem . Let & be the formal differential operator of that theorem . Let 2 be the smallest number in σ ( 7 ) . Let τ have na ( = 0 or 2 ...
... preceding lemma . Q.E.D. Next we turn to the more detailed analysis of case ( b ) of the preceding theorem . Let & be the formal differential operator of that theorem . Let 2 be the smallest number in σ ( 7 ) . Let τ have na ( = 0 or 2 ...
Page 1474
... preceding lemma that if 21 , 22 € J , and λ1 << λ2 , then J. Thus J is an interval . Our second assertion follows immediately from the preceding lemma . Q.E.D. 45 LEMMA . At most one eigenvalue of T lies in any interval Jn · PROOF ...
... preceding lemma that if 21 , 22 € J , and λ1 << λ2 , then J. Thus J is an interval . Our second assertion follows immediately from the preceding lemma . Q.E.D. 45 LEMMA . At most one eigenvalue of T lies in any interval Jn · PROOF ...
Page 1771
... preceding theorem and Theorem 6.23 . Statement ( ii ) follows from statement ( ii ) of the preceding theorem , since a function satisfying the hypotheses of the present statement ( ii ) evidently ( cf. Theorem 6.23 ) satisfies the ...
... preceding theorem and Theorem 6.23 . Statement ( ii ) follows from statement ( ii ) of the preceding theorem , since a function satisfying the hypotheses of the present statement ( ii ) evidently ( cf. Theorem 6.23 ) satisfies the ...
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