Linear Operators, Part 2 |
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Page 1378
... preceding theorem . Moreover , in the course of the proof preceding the statement of Theorem 23 , it was shown that if 2 is any point in A , there exists a small open subinterval N of A , containing 2 , such that the set of restrictions ...
... preceding theorem . Moreover , in the course of the proof preceding the statement of Theorem 23 , it was shown that if 2 is any point in A , there exists a small open subinterval N of A , containing 2 , such that the set of restrictions ...
Page 1474
... preceding lemma that if 21 , 22 J , and λ1 < 2 < λ2 , then λe Jn . Thus Jn 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 J ...
... preceding lemma that if 21 , 22 J , and λ1 < 2 < λ2 , then λe Jn . Thus Jn 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 J ...
Page 1480
... preceding theorem . Q.E.D. = 54 COROLLARY . Under the hypotheses and with the notation of the preceding theorem , T is defined by at most one boundary condition , which is a boundary condition at an end point a of the interval I. Let 22 ...
... preceding theorem . Q.E.D. = 54 COROLLARY . Under the hypotheses and with the notation of the preceding theorem , T is defined by at most one boundary condition , which is a boundary condition at an end point a of the interval I. Let 22 ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 coefficients compact operator complex numbers constant continuous function converges Corollary deficiency indices Definition denote dense 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 linear 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 subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero