Linear Operators: Spectral theory |
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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 , 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 || T ...
... Hilbert- Schmidt operators may be defined as follows . 1 DEFINITION . Let { x , 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 || T ...
Page 1132
... operator K * is represented by the set of kernels K * ( s , t ) = Kj¿ ( 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 ) = Kj¿ ( 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 | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero