Linear Operators: Spectral theory |
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Page 884
... proved by Gelfand and Nai- mark [ 1 ] . In their proof , they proved Lemma 3.5 by using a fairly deep result of Šilov that was not generally available . The proof of this lemma given here is that given by Arens [ 6 ] , who has also ...
... proved by Gelfand and Nai- mark [ 1 ] . In their proof , they proved Lemma 3.5 by using a fairly deep result of Šilov that was not generally available . The proof of this lemma given here is that given by Arens [ 6 ] , who has also ...
Page 964
... proved , we obtain ( TE ( e ) g , Tg1 ) = ( E ( e ) g , g1 ) ( E ( e ) g , 81 ) = √ . ( Tg ) ( m ) ( Tg1 ) ( m ) μ ( dm ) . * : 0 Since this identity is valid for any g1 in L2 ( R ) , we conclude that T ( E ( e ) g ) ( m ) = Tg ( m ) ...
... proved , we obtain ( TE ( e ) g , Tg1 ) = ( E ( e ) g , g1 ) ( E ( e ) g , 81 ) = √ . ( Tg ) ( m ) ( Tg1 ) ( m ) μ ( dm ) . * : 0 Since this identity is valid for any g1 in L2 ( R ) , we conclude that T ( E ( e ) g ) ( m ) = Tg ( m ) ...
Page 1133
... proved . Q.E.D. We have also proved the following corollary . 6 COROLLARY . The adjoint operator of the operator K of the preceding lemma is defined by the set K of kernels defined by ij K ( s , t ) = K ( t , 8 ) . The following theorem ...
... proved . Q.E.D. We have also proved the following corollary . 6 COROLLARY . The adjoint operator of the operator K of the preceding lemma is defined by the set K of kernels defined by ij K ( s , t ) = K ( t , 8 ) . The following theorem ...
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