Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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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 ( 7E ( e ) g , 7g1 ) = ( E ( e ) g , g1 ) = √ ̧ ( tg ) ( m ) ( Tg1 ) ( m ) μ ( dm ) . Since this identity is valid for any g1 in L2 ( R ) , we conclude that T ( E ( e ) g ) ( m ) = g ( m ) for u - almost all m in e ...
... proved , we obtain ( 7E ( e ) g , 7g1 ) = ( E ( e ) g , g1 ) = √ ̧ ( tg ) ( m ) ( Tg1 ) ( m ) μ ( dm ) . Since this identity is valid for any g1 in L2 ( R ) , we conclude that T ( E ( e ) g ) ( m ) = g ( m ) for u - almost all m in e ...
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 K ( s , t ) ij = K¡¡ ( t , s ) . The following ...
... 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 K ( s , t ) ij = K¡¡ ( t , s ) . The following ...
Contents
IX | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients compact subset complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain eigenvalues element essential spectrum 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(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood norm open set open subset orthonormal partial differential operator Plancherel's theorem positive PROOF prove real axis real numbers satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose T₁ T₁(t theory To(t topology unique unitary vanishes vector zero