Linear Operators: Spectral theory |
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Page 1009
... to work with an abstract Hilbert space rather than one of its rep- resentations as an L - space and in this setting XI.6 1009 HILBERT - SCHMIDT OPERATORS Hilbert-Schmidt Operators Unbounded Operators in Hilbert Space 1 Introduction.
... to work with an abstract Hilbert space rather than one of its rep- resentations as an L - space and in this setting XI.6 1009 HILBERT - SCHMIDT OPERATORS Hilbert-Schmidt Operators Unbounded Operators in Hilbert Space 1 Introduction.
Page 1010
... Hilbert- Schmidt operators may be defined as follows . 1 DEFINITION . Let { xa , α = 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 { xa , α = 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 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 ) ...
... 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 ) ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
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 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero