Linear Operators: Spectral theory |
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Page 1130
... kernel K ( a , b ) , then the adjoint operator K * is represented by the kernel K ( b , a ) . Conversely , if K ( · , · ) is a kernel satisfying ( 1 ) , then ( 2 ) defines a Hilbert - Schmidt operator . To prove these assertions , let ...
... kernel K ( a , b ) , then the adjoint operator K * is represented by the kernel K ( b , a ) . Conversely , if K ( · , · ) is a kernel satisfying ( 1 ) , then ( 2 ) defines a Hilbert - Schmidt operator . To prove these assertions , let ...
Page 1131
... kernel representing K * is the function K ( b , a ) = K * ( a , b ) with the Fourier coefficients C If K ( a , b ) is a kernel satisfying the inequality in ( 1 ) , then we have seen that the right side of ( 2 ) defines a bounded ...
... kernel representing K * is the function K ( b , a ) = K * ( a , b ) with the Fourier coefficients C If K ( a , b ) is a kernel satisfying the inequality in ( 1 ) , then we have seen that the right side of ( 2 ) defines a bounded ...
Page 1624
... kernel of Volterra type : ( * ) f ( t , λ ) = cos t√λ + K1 ( t , 8 ) cos s√āds . Let us indicate briefly how the kernel K1 is obtained once the func- tions f ( t , 2 ) are known . A formal differentiation gives the following partial ...
... kernel of Volterra type : ( * ) f ( t , λ ) = cos t√λ + K1 ( t , 8 ) cos s√āds . Let us indicate briefly how the kernel K1 is obtained once the func- tions f ( t , 2 ) are known . A formal differentiation gives the following partial ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero