Linear Operators: Spectral theory |
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Page 1211
... Hypothesis 7 is satisfied for such differential operators . 9 LEMMA . Under Hypothesis 7 , there is , for each g in La ( S , E , v ) , a function W defined on the Cartesian product of S and the real number system R which is measurable ...
... Hypothesis 7 is satisfied for such differential operators . 9 LEMMA . Under Hypothesis 7 , there is , for each g in La ( S , E , v ) , a function W defined on the Cartesian product of S and the real number system R which is measurable ...
Page 1215
... hypothesis of the preceding theorem we have f ( s ) = Σ √ ° ( Uƒ ) , ( A ) W ( s , 2 ) μ ( d2 ) , aε A 81 ƒ Є L2 ( S , Σ , v ) , the integrals existing in the mean square sense in L2 ( S , Σ , v ) and the series converging in the norm ...
... hypothesis of the preceding theorem we have f ( s ) = Σ √ ° ( Uƒ ) , ( A ) W ( s , 2 ) μ ( d2 ) , aε A 81 ƒ Є L2 ( S , Σ , v ) , the integrals existing in the mean square sense in L2 ( S , Σ , v ) and the series converging in the norm ...
Page 1734
... hypothesis , no point in V but the points ( Σ 。~ U1 ) belong to the boundary of q ( I □ U1 ) , and no point in E。 is interior to the closure of I. It follows that ( IU1 ) must consist of one or another of the hemispheres V + { x = V ...
... hypothesis , no point in V but the points ( Σ 。~ U1 ) belong to the boundary of q ( I □ U1 ) , and no point in E。 is interior to the closure of I. It follows that ( IU1 ) must consist of one or another of the hemispheres V + { x = V ...
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 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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero