Linear Operators: Spectral theory |
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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 1⁄4 , such that the set of ...
... 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 1⁄4 , such that the set of ...
Page 1419
... preceding lemma , −f ( t ) ≤ f1 ( t ) in [ 8 ; +1 , mi + 1 ] . In particular -f ( mi + 1 ) = f ( m ; +1 ) ≤ f1 ... preceding corollary . Q.E.D. PROOF OF THEOREM 24. If the function q of Theorem 24 is not negative for t sufficiently ...
... preceding lemma , −f ( t ) ≤ f1 ( t ) in [ 8 ; +1 , mi + 1 ] . In particular -f ( mi + 1 ) = f ( m ; +1 ) ≤ f1 ... preceding corollary . Q.E.D. PROOF OF THEOREM 24. If the function q of Theorem 24 is not negative for t sufficiently ...
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 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 Haar measure 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 operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF 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