Linear Operators: Spectral theory |
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Page 907
... non - negative then , by Corollary 2.7 , ( Ax , x ) = So ( 4 ) λ ( E ( dλ ) x , x ) , which shows that the operator A is positive . Conversely , if an open interval d of negative numbers inter- sects the spectrum then , by Lemma 3.3 ( i ) ...
... non - negative then , by Corollary 2.7 , ( Ax , x ) = So ( 4 ) λ ( E ( dλ ) x , x ) , which shows that the operator A is positive . Conversely , if an open interval d of negative numbers inter- sects the spectrum then , by Lemma 3.3 ( i ) ...
Page 1088
... non - negative . Show that K ( s , t ) = - Σμιφε ( 8 ) φ ; ( t ) , the series converging uniformly . ( Hint : Show ... non - negative self adjoint operator T * T is compact ( Corollary VI.5.5 ) ; thus , by Corollary X.3.5 , Corollary VI ...
... non - negative . Show that K ( s , t ) = - Σμιφε ( 8 ) φ ; ( t ) , the series converging uniformly . ( Hint : Show ... non - negative self adjoint operator T * T is compact ( Corollary VI.5.5 ) ; thus , by Corollary X.3.5 , Corollary VI ...
Page 1254
... non- negative symmetric operator in . By Theorem 5.2 it has a non- negative self adjoint extension S1 . If E ( - ) denotes the resolution of the identity of S1 , it follows from the proof of Theorem 1 that m1 = [ °° _t " μ ( dt ) ...
... non- negative symmetric operator in . By Theorem 5.2 it has a non- negative self adjoint extension S1 . If E ( - ) denotes the resolution of the identity of S1 , it follows from the proof of Theorem 1 that m1 = [ °° _t " μ ( dt ) ...
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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