Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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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 { x , xe 4 } be a complete orthonormal set in the Hilbert space . A bounded linear operator T is said to be a Hilbert - Schmidt operator in case the quantity || T ...
... Hilbert- Schmidt operators may be defined as follows . 1 DEFINITION . Let { x , xe 4 } be a complete orthonormal set in the Hilbert space . A bounded linear operator T is said to be a Hilbert - Schmidt operator in case the quantity || T ...
Page 1132
... operator K * is represented by the set of kernels K ( s , t ) = K ( 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 ( A ) satisfying ...
... operator K * is represented by the set of kernels K ( s , t ) = K ( 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 ( A ) satisfying ...
Contents
IX | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
Copyright | |
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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