Linear Operators: Spectral theory |
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Page 1010
... Hilbert- Schmidt operators may be defined as follows . 1 DEFINITION . Let { x , a E A } 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 ...
... Hilbert- Schmidt operators may be defined as follows . 1 DEFINITION . Let { x , a E A } 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 ...
Page 1013
... operator topology of the sequence { T } . It follows from Lemma VI.5.3 that T is compact . Q.E.D. Not every compact operator is in HS , however . For example , if { } is an orthonormal set in a separable ... HILBERT - SCHMIDT OPERATORS.
... operator topology of the sequence { T } . It follows from Lemma VI.5.3 that T is compact . Q.E.D. Not every compact operator is in HS , however . For example , if { } is an orthonormal set in a separable ... HILBERT - SCHMIDT OPERATORS.
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 ( 4 ) 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 ( 4 ) satisfying ...
Contents
BAlgebras | 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 complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero