Linear Operators, Part 2 |
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Page 1009
... Hilbert - Schmidt Operators In this section the theory of operators of the Hilbert - Schmidt type will be developed and rather deep and fundamental com- pleteness theorems for the eigenfunctions of such operators and associated unbounded ...
... Hilbert - Schmidt Operators In this section the theory of operators of the Hilbert - Schmidt type will be developed and rather deep and fundamental com- pleteness theorems for the eigenfunctions of such operators and associated unbounded ...
Page 1010
... Hilbert- Schmidt operators may be defined as follows . 1 DEFINITION . Let { xa , a € A } be a complete orthonormal set in the Hilbert space H. A bounded linear operator T is said to be a Hilbert - Schmidt operator in case the quantity ...
... Hilbert- Schmidt operators may be defined as follows . 1 DEFINITION . Let { xa , a € A } be a complete orthonormal set in the Hilbert space H. A bounded linear operator T is said to be a Hilbert - Schmidt operator in case the quantity ...
Page 1132
... operator K * is represented by the set of kernels K * ( s , t ) = Kjż ( t , s ) . Finally , if K is any set of kernels satisfying the inequality in ( iv ) , then ( 3 ) and ( 4 ) define a Hilbert - Schmidt operator K in L2 ( 4 ) ...
... operator K * is represented by the set of kernels K * ( s , t ) = Kjż ( t , s ) . Finally , if K is any set of kernels satisfying the inequality in ( iv ) , then ( 3 ) and ( 4 ) define a Hilbert - Schmidt operator K in L2 ( 4 ) ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 coefficients compact operator complex numbers constant 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 subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero