Linear Operators: Spectral theory |
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Page 1653
... Statement ( i ) follows from statement ( ii ) by Definitions 15 ( iii ) and 17 ( ii ) . Statement ( iii ) follows from statement ( ii ) and the fact that F ( x + 1 ) F ( x ) for all k ≥0 and F in H ( +1 ) ( I ) , ( cf. Definition 15 ...
... Statement ( i ) follows from statement ( ii ) by Definitions 15 ( iii ) and 17 ( ii ) . Statement ( iii ) follows from statement ( ii ) and the fact that F ( x + 1 ) F ( x ) for all k ≥0 and F in H ( +1 ) ( I ) , ( cf. Definition 15 ...
Page 1756
... statement ( i ) . Moreover , statement ( i ) enables us to reduce the proof of the existence of the function V to the proof of the following statement . ( ii ) For each r > 0 and p ≥1 , there exists a function V ? in Ĉ ( E " ) , such ...
... statement ( i ) . Moreover , statement ( i ) enables us to reduce the proof of the existence of the function V to the proof of the following statement . ( ii ) For each r > 0 and p ≥1 , there exists a function V ? in Ĉ ( E " ) , such ...
Page 1771
... Statement ( i ) follows from 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 ...
... Statement ( i ) follows from 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 ...
Contents
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